N Schneider
email: nschneid@ph1.uni-koeln.de

Sylvain Bontemps

Sarah Fechtenbaum

Fabrice Herpin

Pascal Tremblin

F Motte

V Ossenkopf

R S Klessen

R Simon

T Csengeri

P C Myers

T Hill

M Cunningham

C Federrath

Understanding star formation in molecular clouds III. Probability distribution functions of molecular lines in Cygnus X

Keywords: ISM: clouds, ISM: extinction, ISM: abundances, ISM: molecules, ISM: structure

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Introduction

Probability distribution functions (PDFs) form the basis of many modern theories of star formation (e.g. Krumholz & McKee 2005;Hennebelle & Chabrier 2008;Federrath et al. 2008Federrath et al. , 2010;;Padoan & Nordlund 2011;Federrath & Klessen 2012, 2013;Hopkins 2013), and are frequently used to characterise properties of the interstellar medium in simulations (Klessen 2000;Vazquez-Semadeni & Garcia 2001;Burkhart et al. 2013, Ward et al. 2014). In summary, a PDF is defined as the probability of finding gas within a column-density 1 range [N, N+dN]. We define η ≡ ln(N/ N ), and the quantity p η (η) then corresponds to the PDF of η. For more details on the basic definitions, see Schneider et al. (2015a).

Observationally, PDFs were first obtained from near-IR extinction maps (Lombardi et al. 2008, Kainulainen et al. 2009, Froebrich & Rowles 2010). With the advent of Herschel2 , it is now possible to determine dust column-density maps that cover a very large dynamic range from A v <1 up to a few hundred A v . Studies using this type of map reveal large variations in dust PDFs. Earlier findings (Kainulainen et al. 2009) of a simple lognormal plus power-law tail distribution for star-forming regions are confirmed (e.g. Schneider et al. 2013Schneider et al. , 2015a;;Stutz & Kainulainen 2015), but more complex shapes are also detected (Hill et al. 2011;Schneider et al. 2012Schneider et al. , 2015c;;Russeil et al. Fig. 2. Line integrated (v=-10 to 20 km s -1 ) maps (in K kms -1 ) of molecular emission in Cygnus X North. The lower level is zero for all maps. The blue crosses in the N 2 H + map indicate submm-continuum sources (Motte et al. 2007).

Fig. 1. 12 CO 1→0 contours of line integrated (v=-10 to 20 km s -1 ) emission with levels 63.5 to 381 K kms -1 in steps of 63.5 K kms -1 overlaid on a Herschel dust column-density map at 36 resolution (that corresponds to 0.41 pc at a distance of 1.4 kpc). This map was corrected for an average foreground contamination of A v = 5. Prominent features, such as W75N or the DR21 ridge, are indicated in the plot. DR17, 22, and 23 are H II regions.

2013; Alves de Oliveira 2014; Tremblin et al. 2014). The outcome of these works is that (i) all clouds have a distribution for low A v that is best described by a lognormal, (ii) a second peak at low A v can emerge in the case of external pressure, which represents the compressed shell, and (iii) a single or double powerlaw tail is found for star-forming regions, and it is proposed that this is related to gravitational collapse and/or external pressure (either as a phase transition between clumps and interclump gas or stellar feedback).

Only a few studies attempt to make PDFs from molecular line observations. N(H 2 )-PDFs of 12 CO or 13 CO (Goldsmith et al. 2008, Wong et al. 2008, Goodman et al. 2009, Lo et al. 2009, Carlhoff et al. 2013, Schneider et al. 2015b) are often clipped at a certain A v threshold because the lines can become optically thick and CO is depleted. Taking W43 as an example, this threshold for 13 CO 2→1 lies between A v = 30 (for an assumed T ex =10 K) and ∼100 (for T ex =12 K and correction for high opacity). Assembling a PDF from one or several molecular line tracers with high critical densities such as CS, HCO + , HCN, N 2 H + etc.

has not yet been tried owing to the lack of extended maps. In this study, we use our large dataset of Cygnus X in the molecular lines of 12 CO, 13 CO, and C 18 O 1→0, CS 2→1, and N 2 H + 1→0 that were obtained with the Five Colleges Radio Astronomy Observatory (FCRAO), and dust column-density maps of the Cygnus X North region (Hennemann et al. 2012, Schneider et al. 2015d) to produce PDFs from H 2 column density that were obtained from molecules and dust. We discuss their properties but also point out the large uncertainties and observational difficulties related to molecular line PDFs.

Observations

Molecular line data

We use data from a molecular line survey of 12 CO (115.3 GHz), 13 CO (110.2 GHz), C 18 O 1→0 (109.8 GHz), CS 2→1 (98.0 GHz), and N 2 H + 1→0 (93.2 GHz), which were collected with the FCRAO 14m radiotelescope between 2003 and 2006. The data were obtained using the 32 pixel Second Quabbin Optical Imaging Array (SEQUOIA) in an on-the-fly (OTF) observing mode. The beamwidth of the FCRAO at 93 GHz is ∼48 and at 110 GHz ∼45 , while the main beam efficiency at that time was 0.48. For the present paper, we use maps on a 20 grid. The data have a mean 1 σ rms rms noise level of ∼0.4 K per 0.06 km s -1 channel on a T mb temperature scale that we use in this paper. In the C 18 O map, the noise level and a slightly stripy appearance, resulting from the on-the-fly mapping mode, becomes apparent. However, this does not affect the PDFs because they are assembled from pixels above the 3σ rms level. For further details, see Schneider et al. (2010Schneider et al. ( , 2011)).

Column-density maps from Herschel

For this study, we employ a column-density map at 36 angular resolution (regridded to 20 to match the fully-sampled FCRAO maps) that were obtained as part of the HOBYS (Herschel imaging survey of OB Young Stellar objects) key program (Motte et al. 2010). This map was obtained by an SED (Spectral Energy Distribution) fit to the PACS (Poglitsch et al. 2010) 160 µm, and the SPIRE (Griffin et al. 2010) 250, 350, and 500 µm wavelengths observations. More details can be found in Appendix A and a full map of the column density of the Cygnus X North region is shown in Schneider et al. (2015d) and Bontemps et al. (in prep.). A smaller cutout of the DR21 ridge at 25 resolution is presented in Hennemann et al. (2012).

The structure of the Cygnus X North region

Cygnus X is one of the richest star-formation regions in the Galaxy, containing the prominent OB-association Cyg OB2, with ∼50 O-stars (see [START_REF] Reipurth | Handbook of star forming regions[END_REF] for a review). From large-scale 13 CO 2→1 (Schneider et al. 2006) and 13 CO 1→0 (Schneider et al. 2007) surveys, we derive a mass of a few 10 6 M for the whole molecular cloud complex that is divided into the Cygnus X North and South regions. These studies show that the majority of the molecular clouds in the complex are located at a common distance of about 1.4-1.7 kpc. A distance of 1.4 kpc, which is derived from maser parallax (Rygl et al. 2012), is now commonly accepted. The clouds in the Cygnus X North region are actively forming stars and contain more than 100 massive pre-and protostellar dense cores (Motte et al. 2007, Bontemps 2010, Csengeri et al. 2011, Hennemann et al. 2012).

Figure 1 shows the dust column-density map of Cygnus X North with contours of 12 CO 1→0 overlaid onto it. The 12 CO emission is integrated over the velocity range -10 to 20 km s -1 , which confines all star-forming molecular clouds that are directly associated with the Cyg OB2 cluster. Outside this velocity range, there are no other clouds along the line of sight, although in other regions of Cygnus X, clouds from the Perseus arm appear at velocities around -40 km s -1 . As demonstrated in Schneider et al. (2006, see their Fig. 3), there are mainly two velocity coherent cloud complexes in Cygnus X North: the DR21 ridge-DR22-DR23 clouds at v=-10 to 1 km s -1 and the W75N-DR17 clouds at v=7 to 20 km s -1 . However, 12 CO and dust also trace more diffuse emission, mainly arising from the 'Great Cygnus Rift', a nearby (0.6-1 kpc) region of obscuration, which is identified in optical images, and which is not associated with Cyg OB2. As shown in Schneider et al. (2007), the extinction due to the Rift is of the order of A v ∼5-10 and mainly arises between v=6 to 20 km s -1 . Other authors determine values between A v =2-5 (Sale et al. 2009), A v =2.5-7 (Drew et al. 2008), A v =5.5-7.5 (Wright et al. 2010), and A v =2.6-5.6 (Guarcello et al. 2012). An extinction around five is also the lowest emission level in the Herschel dust map that is analysed in this paper. The Rift emission is barely detected in 13 CO and not in C 18 O, CS, and N 2 H + . This means that we expect only the low column-density range of the 12 CO PDF to be affected. The higher column-density power-law tail is, in any case, not affected.

Figure 2 shows the same cutout of Cygnus X North as Fig. 1 in four different line tracers. The optical depth of the lines decreases from left to right and the critical density increases. While the CO 1→0 lines have a low critical density n cr of ∼10 3 cm -3 , the CS 2→1 and the N 2 H + 1→0 lines have n cr of 1.3×10 5 cm -3 and 6.1×10 4 cm -3 , respectively, for a temperature of 10 K and thermal excitation (see. e.g. Flower 1999, Daniel et al. 2005, Fig. 3. PDF derived from the foreground-corrected Herschel column-density map. The left y-axis gives the normalized probability density p(η), the right y-axis the number of pixels per log bin. The upper x-axis is the visual extinction and the lower xaxis the natural logarithm of the normalized column density. The green curve indicates the best lognormal fit to the low columndensity distribution, the red line displays a linear regression power-law fit to the high column-density tail. The dispersion of the fitted PDF (σ η ), the slope s and its error, and the exponent α of an equivalent spherical density distribution (C) and a cylindrical density profile, such as for filaments (F), are indicated. Fig. 4. The original PDF of a model cloud (see text) with a lognormal core and a power-law tail is shown with a green dashed line. This cloud is 'contaminated' with a model cloud that has a lognormal core. The resulting PDF is shown as a blue dashed line. The red line then displays the 'corrected' PDF if a constant value of A v =5 is removed. Shirley et al. 2015). In addition, CO depletes in cold and dense gas while CS, and in particular N 2 H + , remain stable in this gas phase. Accordingly, the maps show selectively different density regimes of the gas. While 13 CO 1→0 is still sensitive to lowerdensity gas and resembles the 12 CO map, N 2 H + is confined to the densest clumps. Basically all clumps visible in the N 2 H + map contain cold, dense cores (starless and protostellar, indicated by crosses in Fig. 2) observed in mm-dust continuum with the MAMBO bolometer (Motte et al. 2007). From the submmdust observations, a typical size of 0.1 pc (0.7 pc) and density of 10 5 cm -3 (10 4 cm -3 ) for the cores (clumps) was derived. This value is confirmed by a decomposition of our N 2 H + map using the GAUSSCLUMPS algorithm (Stutzki & Güsten 1990) in the velocity range -10 to 20 km s -1 , which shows an average clump size of ∼0.6 pc. We only consider clumps larger than 1.5 times the beamsize (45 ). Fig. 5. Probability distribution functions of H 2 , obtained from dust and 12 CO, 13 CO, and C 18 O 1→0. All PDFs were constructed using pixels above the 3σ noise level.

Probability distribution functions from dust and molecules

The dust PDF

As discussed in the previous section, the dust map may suffer from foreground contamination that mainly arises from the Cygnus Rift. Contamination always modifies the measured cloud PDF. Relative to the underlying PDF, the measured PDF becomes narrower, the peak shifts towards higher A v , and the slope of the power-law tail steepens (Schneider et al. 2015a).

Following the method outlined in Schneider et al. (2015a), we can correct for foreground contamination by removing the constant A v =5 value for the Cygnus Rift from the original columndensity map.

For a more realistic approach, we have to take into acount, however, that the foreground contamination will typically also show a distribution of column densities, usually also following a log-normal distribution. In a systematic study, Ossenkopf et al. (in prep.) show that the correction by a constant foreground subtraction still works quite well if the most probable contaminating column density is used as the offset. This is illustrated in Fig. 4. Following the description in Schneider et al. (2015a), we contaminate a model cloud with the parameters of the Cygnus PDF with a foreground cloud that is generated from a fractional Brownian motion (fBm) map. This has a power spectral index of 2.8, which matches the typically-observed spatial scaling behaviour, and has a lognormal column-density PDF with a width σ η = 0.45 and a contamination A v = 5.0. This cloud is thus a more realistic representation of a foreground or background contaminating cloud. Figure 4 shows that the contamination produces a distortion in the PDF but the simple correction obtained by subtracting a constant A v =5.0 value fully recovers the PDF tail and recovers the PDF peak position approximately.

Even for such a relatively strong contamination, we can thus recover the main PDF properties by subtracting a constant offset for the contamination. Figure 3 shows the N-PDF obtained from Herschel dust continuum data, corrected for a contamination of A v =5. This dust PDF shows the typical shape of PDFs obtained for star-forming regions: a lognormal distribution for low column densities followed by a power-law distribution between A v ∼10-15 and a few hundred A v . Above A v ∼100, some excess in the PDF is observed that could constitute a second power law, when compared to a simple power law. Such an excess/flatter power-law tail was recently found in some massive GMCs (Schneider et al. 2015c) and was interpreted as a possible result of internal stellar feedback. The same argument may hold here because only pixels in the DR21 ridge, W75N, and some small, isolated clumps -all associated with young stellar objects (YSOs) -contribute to this high column-density part of the PDF (see Fig. 1). Fig. 7. N-Probability distribution functions obtained from CS using (i) different excitation temperatures (5, 10, 15 K) and an abundance of 4×10 -10 , and (ii) at a temperature of 10 K but abundance values of 10 -10 and 10 -9 . The dust PDF is shown for comparison.

The PDFs from CO

The PDFs of H 2 column densities N mol (H 2 ), which were obtained from molecular lines, are shown in Fig. 5 and6, together with the same dust PDF from Fig. 3. Appendix B explains in detail how N mol (H 2 ) was derived. In summary, we assume a constant excitation temperature of 10 K, LTE conditions, and a beam/source-filling factor of unity when determining all column densities. We note, however, that PDFs are always resolutionlimited. The PDFs were constructed from pixels above the 3σ noise level and are normalized to the number of pixels from the dust map (#pix dust ), i.e. we scaled the molecular line PDFs by the factor #pix mol /#pix dust . In Appendix C, we show the individual PDFs without this scaling. Above the noise level of A v ∼1-2, the N mol (H 2 )-PDFs of 12 CO, 13 CO, and C 18 O can be fitted by a broad lognormal distribution with a peak at A v ∼3-5, and by a power-law tail above A v =15 for C 18 O. Also, the 13 CO PDF shows some excess above A v ∼15 (see Appendix C for a more detailed display).

Considering all uncertainties in the determination of H 2 column density from dust and CO, the lognormal part in the CO-PDFs corresponds quite well to the lognormal part of the dust PDF, which indicates that the low-density molecular gas is probably well mixed with dust. The 12 CO PDF is a special case because it is contaminated by line-of-sight confusion from the Cygnus Rift at a distance of 600 pc, and thus shifts towards lower A v to some degree. For A v larger than ∼15 (we note that the column densities are correct only within a factor ∼2), however the 12 CO PDFs depart from the lognormal shape and the distribution becomes 'bumpy' and falls off. This is the regime where the line no longer traces the dense gas because it becomes optically thick. In comparison to hydrodynamic simulations with radiative transfer (Shetty et al. 2011), our observed 12 CO PDF covers a lower column-density range. Their 'Milky Way cloud' (case (a) in Fig. 2) 12 CO PDF has the same width but peaks around A v =10-15. More recent results of Szücs et al. (2015), however, show 12 CO PDFs that cover a very similar column-density range to the one we observed.

The 13 CO and C 18 O PDFs extend to higher A v (up to ∼40). The 13 CO line is optically thin (τ ∼0.1-0.3, see Fig. B.3) in most of the clouds and becomes marginally more optically thick (τ ∼1) only in the DR21 ridge. The C 18 O line is optically thin everywhere with maximum values of τ ∼0.4 in the DR21 ridge. Optical depth effects, and not depletion, are the main reason for the different PDF shapes of 12 CO, 13 CO, and C 18 O for higher column densities. The power-law behaviour for A v >15 that we find for C 18 O indicates that the whole column of gas is traced up to A v ∼40. Because 13 CO becomes marginally optically thin, we do not observe a clear power-law tail but only some excess because the highest column densities are no longer traced. 12 CO becomes even more easily optically thick so that the high column-density pixels are not traced at all and the PDF appears lognormal. Depletion of all CO isotopologues probably sets in for A v larger than around 40 where all PDFs fall off. The removal of CO from the gas phase typically happens at densities above 10 4 cm -3 at temperatures ∼15 K (e.g. Blake et al. 1995). Our value of A v ∼40 (N=3.8×10 22 cm -2 ) thus corresponds to a typical clump size d of ∼1.25 pc (d = N/(10 4 cm -3 )) in which CO depletes. These clumps are the ones that are well traced by N 2 H + emission and the temperature map (Fig. 8) that we discuss in the next section.

At the low A v end, approximately in the range A v =1-2, we observe some excess compared to a purely lognormal PDF distribution for 13 CO and C 18 O (Fig. C.1). The chemistry in this column-density regime is governed by (F)UV radiation via photodissociation (A v <5) and fractionation (A v <3) processes (e.g. van Dishoeck & Black 1988). The excess indicates that selfshielding of 13 CO and C 18 O against UV-radiation is rather effective. Otherwise, the abundances would be lower and the PDF would drop more significantly. On the other hand, in warmer regions (internally or externally heated) in the medium A v range, CO desorption leads to a higher abundance of CO. These spatial abundance variations, which were observed for 13 CO in Perseus (Pineda et al. 2008) and Orion [START_REF] Ripple | [END_REF], have an obvious influence on the N-PDF. However, they are difficult to assess (see next Section).

The PDFs from CS and N 2 H +

The N mol (H 2 )-PDFs of CS and N 2 H + mostly cover the high column-density range and can be best fitted by a pure power law (see Appendix C). However, all absolute values need to be taken with caution because they strongly depend on the abundance values that were used for converting column densities into H 2 and on the excitation temperature.

For N 2 H + , we adopt [N 2 H + ]/[H 2 ] = 5×10 -10 , a value determined for massive 3 cloud cores (Pigorov et al. 2003), and applied it to one of the Cygnus X North cores (N63, Motte et al. 2007) in the following way (Fechtenbaum et al., in prep.): Level population diagrams that use lines of the optically thin isotopologues 15 N 2 H + and N 15 NH + were plotted to estimate their column density. An isotopic ratio 14 N/ 15 N of 448 (Wilson & Rood 1994) was used to derive the column density of N 2 H + . The mass of 44 M in a 30 beam (Duarte-Cabral et al. 2013) of the source enabled us to calculate the column density of H 2 and thus the abundance of N 2 H + . The resulting PDF covers a column-density regime of A v ∼15 to 400, which corresponds very well to the one derived from dust (see Fig. 6).

For CS, there is a large scatter in observed values for highmass star-forming regions with [CS]/[H 2 ] = 10 -9 to a few times 10 -10 ( Gianetti et al. 2012, Li et al. 2015, Neufeld et al. 2015). Low-mass cores have typical values of a few times 10 -9 (e.g. Tafalla et al. 2006). Here, we use a value of 4×10 -10 (Fechtenbaum et al., in prep.) that was obtained using the optically thin isotopologue C 33 S and applying the same method as outlined above. The PDF obtained with this abundance value covers the A v range ∼20 to 350 and thus has a similar column density range as the PDF determined from dust, but shifted vertically to the latter.

To investigate in more detail the effects of different excitation temperatures and abundances, in Fig. 7 we plot a whole set of PDFs for CS. We first vary the excitation temperature, using 5 and 15 K as two extreme cases, and 10 K (average value determined from all molecular lines and dust). The column density variation is significant, increases its value by more than a factor two between T ex =15 K and 5 K. Different abundance values obviously have a significant impact on the PDF as well. A lower (but unlikely) [CS]/[H 2 ] ratio of 10 -10 shifts the PDF to higher column densities with unrealistic values of a few hundred A v . A higher abundance shifts the PDF towards lower column densities, and a value of 10 -9 leads to a PDF that corresponds to the one obtained from dust. In this way, it is possible to 'calibrate' the [CS]/[H 2 ] abundance using the dust PDF. However, this method should be treated with caution because the molecu- lar line PDF also depends on the excitation temperature, and the dust PDF has its own uncertainties (variable temperature along the line of sight, unknown dust opacity etc.).

The high column density range in the maps

The PDFs obtained from dust emission and N 2 H + (and CS) both show a power-law tail starting around A v ∼15. Figure 8 (left panel) illustrates the high spatial correspondence between the pixels in the dust and N 2 H + maps that cover the same columndensity range. The black and red contours indicate the A v = 15 level for the H 2 column density derived from dust and N 2 H + , respectively. On average, both contours cover the same areas, implying that the dust is well mixed with the dense and cold gas that is traced by N 2 H + . The geometry of the emitting regions is a mixture between spherical and filamentary. Even at very high column densities (indicated by the arbitrary chosen A v =100 blue contour from N 2 H + ), not only spherical clumps are outlined but also the elongated DR21 ridge. An overlay between the dust temperature obtained with Herschel and the same A v =15 contour from N 2 H + (Fig. 8, right panel) shows that it is indeed mostly cold gas traced by N 2 H + and dust. The clumps that are indicated by the red contour typically have a temperature of 10-15 K, except those with internal sources (the DR21 region in the DR21 ridge and DR23 at the southern border of the map). In these cases, the column density is underestimated so that the true power-law tails of both, the dust, and N 2 H + PDF, which is shown in Fig. 6, are slightly flatter. 1. Results from the power-law fit to the PDFs. s and ∆s are the slope and its error, α c and α f are the exponents of a density distribution with ρ ∝ r -α with spherical and filamentary geometry.

Discussion and conclusions

In the following section, we base our discussion on the column density maps and PDFs of CS and N 2 H + derived using an excitation temperature of 10 K and abundance values of [CS]/[H 2 ] = 10 -9 (to be consistent with the dust PDF) and [N 2 H + ]/[H 2 ] = 5×10 -10 , respectively.

Thermal and subthermal excitation regimes

For a homogeneous medium with beam-filling of unity, the H 2 column-density threshold N for thermally excited N 2 H + and CS calculates from the clump size (d) and critical densities with N = d × n cr . For d we adopt 0.7 pc, taken from dust submm observations (Motte et al. 2007). This value is close to the average clump size of 0.6 pc we derive from our N 2 H + clump decomposition 4 . We thus obtain A v =285 and 130 for CS and N 2 H + , respectively, assuming critical densities of 1.3×10 5 cm -3 (6.1×10 4 cm -3 ) for CS (N 2 H + ). Above these values, the emission lines should be thermalized. However, the clumpiness of the gas leads to a lower threshold of thermalization. With an average density n =10 4 cm -3 of the clumps,which was determined from submm continuum observations (Motte et al. 2007), we obtain a beam filling ( n /n cr ) of ∼10% for CS and ∼15% for N 2 H + . Both lines are thus already thermalized at lower A v . For N 2 H + we independently derive its thermal excitation by the hyperfine-structure line ratios. The line is mostly thermally excited because we obtain excitation temperatures above 5 K for the emitting gas (Appendix C), which is above the typical value for subthermal excitation.

CS has a higher critical density but depletes at densities above ∼10 5 cm -3 , which is consistent with our PDF that is cut off at A v ∼100 (for the abundance [CS]/[H 2 ] = 10 -9 ). On the other hand, its chemistry is less density dependent. Hence, we expect that CS is not thermalized at the lowest end of the distribution.

Slopes of the N-PDFs

Independent of the exact column-density regime that is covered by the PDFs from dust and CS/N 2 H + , they all show power-law tails with slopes s between -1.4 and -2.3 (see Table 1). The slope s of the power law can be converted into the exponent α of an equivalent density distribution ρ ∝ r -α (Appendix D). For spherical geometry, representing a single core or a core ensemble (Girichidis et al. 2014), we use α c =1-2/s (Federrath & Klessen 2013). For singular polytropic cylinders, which portray filaments, we use α f =1-1/s (see Toci & Galli (2015) for regular polytropic cylinder models and Fischera (2014) and Myers (2015) for the corresponding N-PDFs). The values we obtain for α (α c or α f ) vary between 1.4 and 2.4. In the simplified picture of the free-fall of a collapsing sphere, α=2 for early stages and α = 1.5 after a singularity formed at the centre of the sphere (Shu 1977, Larson 1969, Penston 1969, Whitworth & Summers 1985). Our observed values from the dust (α c =1.9) and C 18 O (α c =1.8) PDF slopes are consistent with the spherical free-fall scenario, but CS and N 2 H + have higher values with α c =2.3 and α c =2.4, respectively. All high-density clumps traced in CS and N 2 H + emission are associated with pre-and protostellar sources (Motte et al. 2007), which indicates that gravitational collapse in the star-forming phase has started. The high column densities seen in dust, CS, and N 2 H + are thus not a consequence of a long column of diffuse emission, but correspond to spatially concentrated high volume densities.

On the other hand, the purely spherical free-fall picture seems unlikely to apply to most of the gas which defines the N-PDF power laws, because most of the gas we observe in Cygnus X is filamentary. This is clearly illustrated in the maps shown in Figs. 1, 2, and7, which display the filamentary network in the CO lines and, with only a small fraction of gas in cores, that could be considered roughly spherical (see CS and N 2 H + maps in Figs. 2 and7 and submm continuum observations in Motte et al. 2007) 5 . Filaments can be in global free-fall collapse, as suggested by molecular line observations (e.g. Peretto et al. 2013, Schneider et al. 2010, 2015b) but the gas we see in filaments (and cores) can account for power-law N-PDFs if it has the right power-law radial-density structure, even in hydrostatic balance density regime. They are thus more precise for separating individual clumps. 5 In Aquila (Könyves et al. 2015) the power-law tail contains more than ∼50% of the mass in filaments but only ∼15% in cores.

with no net inward motion. The observed N-PDF slopes (s=-1.4 to -2.6, leading to α f between 1.4 and 1.7) are consistent with self-gravitating but non-collapsing filament models (Myers 2015, Toci & Galli 2015). In this case, filamentary gas that has a sufficient mass per length ratio is centrally concentrated and self-gravitating, but does not need to be globally collapsing. However, to confine the filaments/clumps, an external pressure p ext ∝ n T kin of the order of p ext ∼ 10 4-5 cm -3 × 10 K is required (where n is the average density in the filaments/clumps and T kin the gas temperature). However, this pressure cannot be isotropic, provided by the lower density envelope that is well traced in 12 CO at a density of ∼10 3 cm -3 and a temperature of ∼10-40 K. Moreover, it could be ram pressure of new gas that is accreting onto filaments and clumps, a process that is now frequently observed (e.g. Schneider et al. 2010, Kirk et al. 2013, Palmeirim et al. 2013).

The flatter power-law slopes of the CS and N 2 H + PDF in comparison to the dust could reflect an abundance change because N 2 H + is produced when CO is depleted at very high column densities (Bergin et al. 2001, Tafalla et al. 2002, 2006). In addition, because CS is not thermalized at the lowest column densities, this could lead to a somewhat flatter tail with respect to the dust.

In summary, a power-law distribution in the density structure is required to have a power-law tail in the PDF and this can be achieved by either a hydrostatic configuration, where the power law arises from a balance of gravitational forces and pressure gradients, or directly in a dynamically collapsing system. Probably both play a role, with filaments possibly balanced or at least contraction-slowed by (magnetic) pressure gradients on the one hand and very dense collapsing cores on the other.

Summary

We derived probability distribution functions (N-PDFs) of H 2 column density for the Cygnus X North region from dust, 12 CO, 13 CO, C 18 O 1→0, CS 2→1, and N 2 H + 1→0. The determination of the H 2 column density from the molecular lines is based on standard procedures and abundances, assuming a common excitation temperature of 10 K and LTE. Our findings are:

• The PDFs of dust and CO isotopologues can be described by a lognormal distribution for the low column-density range between A v ∼1 to ∼15, with a common peak around A v ∼5. Though line-of-sight contamination and variations in abundance and excitation temperatures introduce large uncertainties in the CO-PDFs, the overall correspondence between these observational PDFs and the ones from recent simulations (Szücs et al. 2015) is good.

• Optical depth effects are the main reason for the different PDF shapes of 12 CO, 13 CO, and C 18 O for higher column densities. The PDF from 12 CO is cut above A v ∼30 because the line becomes optically thick. The marginally optically thin 13 CO line shows excess in its PDF for A v =15-40, while the PDF for the optically thin C 18 O line displays a power-law tail in the same A v =15-40 range.

• Neither selective photodissociation nor fractionation seem to play a significant role for the CO abundances for low column densities.

• Depletion of all CO isotopologues probably sets in for A v greater than around 40 where all PDFs fall off.

• The PDFs of CS and N 2 H + only consist of a power-law tail that covers a high column density range (A v ∼15 to a few 100 A v ). Using CS as an example, we discuss the potential influence of abundance and excitation temperature variations, and show N. Schneider: III. molecular line PDFs that these could shift the entire PDF by more than one magnitude in column density. To better constrain this uncertainty, we 'calibrate' the molecular line PDF using the dust-emission map (i.e. shifting the distribution at a given excitation temperature so that it corresponds to the dust PDF). For CS, we obtain an abundance of [CS]/[H 2 ]=10 -9 in this way.

• We find that the N 2 H + and CS lines are mainly thermally excited. N 2 H + is well mixed with the dust and traces spatially the same dense gas clumps of a typical size of ∼0.6-0.7 pc that were shown to contain pre-and protostellar sources (Motte et al. 2007).

• The slopes of the high column-density power-law tail end of the N-PDFs from dust, CS, and N 2 H + are s=-2.3, -1.6, and -1.4, respectively. These values are consistent with a gas distribution that is dominated by gravity, i.e. free-falling gas in cores and filaments, although a hydrostatic configuration with ram pressure by gas accretion can also take place. The power law then arises from a balance of gravitational forces and pressure gradients.

Summarising the observational results, we find that the N-PDFs obtained from molecular lines are not well confined because they depend strongly on excitation temperature and abundance, and various combinations of these can lead to the same PDF. 'Calibrating' a molecular line PDF with dust is an appealing approach but it should be kept in mind that dust PDFs also suffer from uncertainties. For example, the specific dust opacity is not constant (a value of β=2 was chosen for this paper) and line-of-sight contamination can lead to an overestimation of the column density. However, this study nevertheless shows that the dust provides the highest dynamic range in tracing the low to high column-density regime, compared to the molecular line tracers (see also Goodman et al. 2009 andBurkhart et al. 2013).

B.1. Determination of and constraints on the excitation temperature

The temperature stucture in the Cygnus X North region is complex. Large-scale UV illumination from the central Cyg OB2 cluster heats the diffuse gas and PDR surfaces. Internal (proto)stars and clusters such as DR21 show up as local hotspots. In parallel, the Cygnus X North clouds also contain a large reservoir of cold and dense gas, seen in submm continuum (Motte et al. 2007).

The observations of several molecular lines enable us to trace sensitively this temperature structure. If the LTE assumption is valid and the density is larger than the critical density of the transition, the kinetic temperature of the gas (T kin ) corresponds to the excitation temperature T ex for the molecular lines (which is assumed to be equal for all species). If gas and dust are well mixed, this temperature should also correspond to the dust temperature T dust that was derived from Herschel. However, LTE conditions are not present and the density varies from low values in the diffuse gas phase, mainly seen in 12 CO, to high values in dense clumps in the molecular clouds, traced by dust and the CS and N 2 H + lines. In the following, we determine the excitation temperature in different ways, using the observational data sets.

T ex from 12 CO 1→0 From 12 CO 1→0, we calculate the excitation temperature assuming that this line is optically thick so that T ex ( 12 CO) = 5.53 ln(1 + 5.53/(T mb ( 12 CO) + 0.818))

.

(B.1)

The molecular clouds in the Cygnus region consist of several velocity components (Schneider et al. 2006). We performed a Gaussian line fit to the four main lines (see explanations for the 13 CO column density). Figure B.1 shows the resulting excitation temperature for the most prominent component at velocities around -4 km s -1 . It becomes obvious that the temperature varies significantly between ∼5 and ∼40 K.

T ex from N 2 H + 1→0
We calculate the excitation temperature from N 2 H + by fitting its hyperfine structure with the known relative intensities of the seven components. The fit delivers simultaneously the opacity τ and T ex and shows no anomaly in the relative intensities.

The average value of T ex is 7 K (in a range of 5 to 20 K) and the opacity is generally below one. However, we emphasise that only the product τ×T ex is constrained by the fit so that the excitation temperature is not well constrained.

T ex from dust

The right-hand panel of Fig. 6 shows the dust temperature we derive from the SED fit to the Herschel fluxes. The average value for the dust temperature across the map is 15 K, with a variation between 10 and 25 K (see also Hennemann et al. 2012). Because the dust temperature is an average along the line of sight, it is not well determined in regions with strong temperature gradients. However, for cold, dense clumps and cores that are associated with CS and N 2 H + emission, we expect a much smaller variation in temperature.

In summary, considering the excitation temperatures determined from the molecular lines and the dust, we use 10 K in the following as the best compromise for the determination of column densities. However, we also discuss other methods (using T ex from 12 CO) to obtain the 13 CO column density, and study the influence of different T ex for the N 2 H + column density.

B.2. Column densities from linear molecules

The beam-averaged total column density N of any optically thin molecule can be determined from the observed line-integrated main-beam brightness temperature T mb with

N [cm -2 ] = f (T ex ) T mb [K] dv[kms -1 ] (B.2)
and for linear molecules

f (T ex ) = 3hZ 8π 3 µ 2 J u exp(E up /kT ex ) [1 -exp(-hν/kT ex )](J(T ex ) -J(T BG )) (B.3) and J(T ex ) = hν k(exp(hν/(kT ex ) -1) (B.4)
and J(T BG ) = J(2.7K), and in which h and k denote the Planck and the Boltzman constants, respectively, E up is the energy of the upper level, ν is the frequency [GHz], µ is the dipole moment [Debye], J u is the upper value of the rotational quantum number and T mb dv is the velocity integrated line intensity on a main beam temperature scale.

The first two terms of the rotational partition function for a diatomic linear molecule which is accurate to 1%, compared to the full term for temperatures above 2 K (Mangum & Shirley 2015), are given by

Z = kT ex hB + 1/3, (B.5)
with the rotational constant B expressed to first order as ν=2BJ u .

To determine all H 2 column densities, we apply a correction to the hydrogen mass of a factor of 1.36 to account for helium and other heavy elements. H 2 column density from 13 CO 1-0 The values for the 13 CO 1→0 transition are ν=110.201 GHz, hν/k=5.29 K, µ=0.112 Debye, J u =1, and J(2.7K)=0.868. The temperature-dependent factor f(T ex ) is 3.22, 2.00, 1.62, 1.58, and 1.70×10 15 K -1 (km/s) -1 cm -2 for 5, 7, 10, 15, and 20 K, respectively. As discussed above, we use a constant value of 10 K for T ex . A change in the excitation temperature between 5 and 15 K implies only a 10% variation in the column-density value. We then use an abundance [ 12 C]/[ 13 C]=70, determined from Wilson & Rood (1994). This value corresponds well to the one derived for Orion A (Langer & Penzias 1990). For the [H 2 ]/[ 12 CO] abundance, we use a value of 1.1 10 4 (Pineda et al. 2010, Fontani et al. 2012).

Because it is known that the 13 CO 1→0 line is not always optically thin, an opacity correction τ 13 /(1exp (τ 13 )) for moderate τ can be applied for the column density (Frerking et al. 1982). To verify if this kind of correction is required, we determine the 13 CO opacity by using the excitation temperature T ex that was obtained from 12 CO and the observed 13 CO main-beam brightness temperature T mb ( 13 CO) with τ 13 =ln(1 -T mb ( 13 CO)/(5.29/(exp(5.29/T ex ) -1) -0.868)).

(B.6)

Because the molecular clouds in the Cygnus region consist of several velocity features (Schneider et al. 2006), we perform a Gaussian line fit to four components for 12 CO and 13 CO (see B.1). As shown in Fig. B.2, the dominating emission comes from the component at -4 km s -1 , which is associated with the molecular clouds of DR21, DR22, and DR23. The component at 9 km s -1 becomes important for the W75 region and southwest of the DR21 ridge. Emission at 3 and 15 km s -1 is more diffuse and on a low intensity level. We calculate the 13 CO opacity for all velocity components and find that the line is optically thin everywhere, even for the -4 km s -1 component that is shown in Fig. B.3. We thus did not include an opacity correction for the determination of the 13 CO column density.

The N-PDFs that were determined from the 13 CO column density, using a constant excitation temperature of 10 K or a variable pixel-to-pixel temperature T ex from 12 CO, do not show a significant difference.

H 2 column density from C 18 O 1-0 The values for the C 18 O 1→0 transition are ν=109.782 GHz, hν/k=5.27 K, µ=0.1098 Debye, J u =1, and J(2.7K)=0.872 K. The temperature-dependent factor f(T ex ) is 3.34, 2.08, 1.69, 1.65, and 1.78×10 15 K -1 (km/s) -1 cm -2 for 5, 7, 10, 15, and 20 K, respectively. As discussed in Sec. B.1, we adopt a value of 10 K for T ex . To determine the H 2 column density, we then use a ratio [H 2 ]/[ 12 CO] of 1.1 10 4 (Pineda et al. 2010, Fontani et al. 2012) and an abundance [ 16 O]/[ 18 O] = 531, which was determined from Wilson & Rood (1994), using a distance of 1.4 kpc.

We determine the C 18 O opacity in the same way for the four line components as we did for 13 CO (see above), using the excitation temperature T ex obtained from 12 CO and the observed C 18 O main-beam brightness temperature T mb (C 18 O) with τ 18 =ln(1 -T mb (C 18 O)/(5.27/(exp(5.27/T ex ) -1) -0.87)).

(B.7)

Maps of the resulting C 18 O opacity show that the line is optically thin for all velocity components and reaches its maximum value of τ 18 ∼0.4 only in the DR21 ridge for the -4 km s -1 component.

H 2 column density from 12 CO 1→0 The 12 CO 1→0 line is optically thick so that it should a priori not be a good tracer for the total column density. However, empirical studies (Strong et al. 1988, Bolatto et al. 2013) show that the line still can trace the total column density/mass of a molecular cloud reasonably well with a conversion factor of 2-2.3×10 20 from line integrated CO main-beam brightness temperature into H 2 column density. Here, we adopt a value of 2×10 20 K -1 (km/s) -1 cm -2 .

H 2 column density from CS 2→1

The values for the CS 2→1 transition are ν=98.0 GHz, hν/k=4.70 K, µ=1.95 Debye, J u =2, and J(2.7K)=1.00. The factor f (T ex ) is 16.1, 9.48, 7.35, 7.00, and 7.41×10 12 K -1 (km/s) -1 cm -2 for 5, 7, 10, 15, and 20 K. We assume optically thin CS emission and an abundance [CS]/[H 2 ] of 4×10 -10 (Fechtenbaum et al., in prep.).

H 2 column density from N 2 H + 1→0

The values for the N 2 H + 1→0 transition are ν=93.17 GHz, hν/k=4.47 K, µ=3.40 Debye, J u =1, and J(2.7K)=1.06. The factor f (T ex ) is 3.62, 2.44, 2.10, 2.16, and 2.39×10 12 K -1 (km/s) -1 cm -2 for 5, 7, 10, 15, and 20 K. The total beam averaged N 2 H + column density is calculated using the line-integrated emission from the F=(2,1) J=(1,0) component at a frequency of 93.1737767 GHz and a HFS intensity ratio of 7/27 for that component. We assume an abundance [N 2 H + ]/[H 2 ] of 5×10 -10 .

We emphasise that all H 2 column-density determinations have a large uncertainty, mainly arising from the abundance. The [CO]/[H 2 ] abundance for the CO isotopolgues is probably better constrained because it has been well studied in the literature. However, the variation in excitation temperature along the line of sight is more significant here because the CO lines trace a larger PDFs derived from the molecular line maps. The left y-axis gives the normalized probability p(η), the right y-axis the number of pixels per log bin. The upper x-axis is the visual extinction and the lower x-axis the logarithm of the normalized column density. For the CO lines, we fitted a lognormal to the low column-density range, indicated by a green curve. The red line shows the power-law fit to the high column-density end for C 18 O, CS, and N 2 H + . The respective slopes s (and errors) are given in the panels. For the CO lines, the peak and width of the PDF is indicated with A v peak and σ.

Fig. 6 .

 6 Fig. 6. Like Fig. 4 but showing the CS 2→1 and N 2 H + 1→0 PDFs (converted into H 2 column density) in addition to the dust PDF.

Fig. 8 .

 8 Fig. 8. Left: Foreground-corrected dust column-density map (as shown in Fig. 1) in greyscale. The black contour indicates the A v = 15 level. The red contours show the same A v =15 level from the H 2 column-density map derived from N 2 H + , using an excitation temperature of 10 K and an abundance of 2×10 -9 . Right: Dust temperature map of Cygnus X North from Herschel with the same red A v =15 contour from N 2 H + .

Fig. B. 3 .

 3 Fig. B.3. Map of the 13 CO opacity derived for the velocity component -4 km s -1 . White pixels indicate positions that were not well-fitted (e.g. FWHM too large, intensities too low) and thus attributed a blanking value.

Fig. B. 2 .

 2 Fig. B.2. Maps of the Gaussian fits to the main-beam brightness temperature of 13 CO 1→0 for the four main velocity ranges in Cygnus X North. The most prominent emission comes from the velocity component at -4 km s -1 .

Fig

  Fig. C.1.PDFs derived from the molecular line maps. The left y-axis gives the normalized probability p(η), the right y-axis the number of pixels per log bin. The upper x-axis is the visual extinction and the lower x-axis the logarithm of the normalized column density. For the CO lines, we fitted a lognormal to the low column-density range, indicated by a green curve. The red line shows the power-law fit to the high column-density end for C 18 O, CS, and N 2 H + . The respective slopes s (and errors) are given in the panels. For the CO lines, the peak and width of the PDF is indicated with A v peak and σ.

Herschel is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA.

Lower abundances around 1-3×10 -10 were found in low-mass dense cores and dark clouds (e.g.Caselli et al. 2002, Bergin et al. 2001, Tafalla et al. 2006).

The submm continuum observations have an angular resolution of 11 , compared to the 45 for N 2 H + , and trace a much larger column

Acknowledgements. N.S. and S.B. acknowledge support by the ANR-11-BS56-010 project STARFICH. N.S., V.O., and T.C. acknowledge support from the Deutsche Forschungsgemeinschaft, DFG, through project number 0s 177/2-1 and 177/2-2, and central funds of the DFG-priority program 1573 (ISM-SPP). C.F. acknowledges funding provided by the Australian Research Councils Discovery Projects (grants DP130102078 and DP150104329). R.S.K. acknowledges subsidies from the DFG priority program 1573 (Physics of the Interstellar Medium) and the collaborative research project SFB 881 (The Milky Way System, subprojects B1, B2, and B5).

Appendix A: Determination of H 2 column density from dust

We assume optically thin dust emission at one dust temperature T d for each pixel in the map and fitted a greybody function of the form I ν = B v (T d ) κ Σ to the absolutely calibrated datapoints at 160 µm, 250 µm, 350 µm, and 500 µm. The specific dust opacity per unit mass (dust+gas) is approximated by the power law κ ν = 0.1 (ν/1000GHz) β cm 2 /g with β=2, and the dust temperature and surface density distribution Σ are left as free parameters. The H 2 column density N dust is then calculated from Σ = µ H m H N dust , adopting a mean molecular weight per hydrogen molecule µ H =2.8. For more details, see Hill et al. (2011). We approximate the final uncertainties in the dust column-density maps to be around ∼30-50%, mainly as a result of the uncertainty in the assumed form of the opacity law and possible temperature gradients along the line of sight. range in densities and temperatures than the high-density tracers CS and N 2 H + . For the latter, the constant excitation temperature assumption is most likely more valid, but the abundances of CS and N 2 H + are less well constrained, in particular for massive clumps and cores. In summary, we consider the H 2 column densities from CO to be correct within a factor of two and the ones from CS and N 2 H + within a factor of a few.

Appendix B: Determination of H 2 column density from molecular lines

Appendix C: Individual molecular line PDFs

In the main body of the paper, we used a normalization for the PDF with respect to the size, i.e. pixel number of the dust map. We here show the PDFs obtained with a normalization

Appendix D: Relation between PDF slope and radial density distribution for different geometries

We define the PDF p N as the surface fraction with gas of column density N and the PDF of the natural log of the column density density p η with η = ln(N/ N ), the relation between the two is p η = N p N . The definition of p N gives p N ∝ dA/dN.

Spherical geometry

In the case of spherical geometry, the density radial profile is

Cylindrical geometry

In the case of cylindrical geometry, the density radial profile from the axis of the cylinder is ρ(r c ) ∝ r -α c , hence N ∝ ρ r c ∝ r -α+1 c . However in this case A ∝ z × r c ∝ r c , hence p N ∝ dA/dN ∝ 1/r -α c ∝ N α/(1-α) p η = N p N ∝ N 1/(1-α) .

(D.2)

Note that similar calculations can be found in Federrath & Klessen (2013), Fischera et al. (2014), and Myers (2015).


Oana Silvia

Editors Walter Briec

Charles Horvath
email: horvath@univ-perp.fr

SOME GENERAL PRINCIPLES IN TROPICAL CONVEXITIES

Some general principles in tropical convexities

Introduction

The text is structured along the lines of a short talk given at ETAMM 2016. Some details are given in the first part, which is mainly algebraic or set theoretical; few details are given in the second part where algebra and topology intertwine. The first part revolves maily around the Kakutani Separation Property, also known as the Stone-Kakutani Property or the algebraic Hahn-Banach; going into the analytic or the geometric Hahn-Banach would have taken us too far. The second part revolves around a basic intersection theorem, the theorem of Knaster-Kuratowski-Mazurkiewicz -which is an avatar of Brouwer's Fixed Point Theorem -and some of its consequences, mainly fixed point theorems for single valued or multivalued maps. We refrained from giving the most general statement or the most general proof, when we gave one at all. With the last part we come back to the first part of the paper; the result stated their shows that the Kakutani Separation Prop erty, along with some natural topological properties, like connectedness of the convex sets, are at the heart of the fixed points results -and others too -that were presented in the preceding section. We have not tried to be exhaustiveduality and variational inequalities in tropical convexities are not to be found here -we tried to show on a few basic results why and how things work. Applications are made mainly to the standard tropical convexities, maxplus in R n and maxtimes, also called B-convexity, in R n + . Some infinite dimensional tropical convexities are listed among the examples but they are not studied here. The topological structures of some infinite dimensional spaces naturally related to tropical convexity are studied in [START_REF] Bazylevych | Spaces of idempotent measures of compact metric spaces[END_REF] and [START_REF] Bazylevych | Hyperspaces of max-plus convex subsets of power of the real line[END_REF].

Convexities and convex spaces

[ A ] ] C = A ⇔ A ∈ C. Furthermore, ∀A, B ⊂ X      (1) A ⊂ [ [ A ] ] C (2) A ⊂ B ⇒ [ [ A ] ] C ⊂ [ [ B ] ] C (3) [ [ [ [ A ] ] C ] ] C = [ [ A ] ] C (1)
For all non empty sets S let S be the family of nonempty finite subsets of S. Condition (Conv3) implies that Lemma 2.1.2. For all convex space (X, C) and for all subsets A of X

[ [ A ] ] C = F ∈ A [ [ F ] ] C (2)
Proof. We can assume that A = ∅. The family A of nonempty finite subsets of A is updirected and consequently so is the family of convex sets

[ [ F ] ] C : F ∈ A and therefore F ∈ A [ [ F ] ] C is convex, call this set C. If F ∈ A then F ⊂ [ [ F ] ] C ⊂ [ [ A ] ] C from which C ⊂ [ [ A ] ] C . From A = F ∈ A F we have A ⊂ C and consequently, since C ∈ C, [ [ A ] ] C ⊂ C. We have shown that [ [ A ] ] C = C. 2
Lemma 2.1.3. For all convex space (X, C) and for all subset C of X, C ∈ C if and only if, for all finite subset

F of C, [ [ F ] ] C ⊂ C.
Proof. This is a restatement of Lemma 2. 2

The C-hull of a finite set is a C-polytope. From 1 and 2 we have

∀C ∈ C C = P ⊂C
P is a polytope

P (3)

Conditions 1 say that the application A → [ [ A ]

] C is a Moore closure operator while 2 says that it is an algebraic closure operator. One can easily check, that given a algebraic Moore closure operator

[ [ • ] ] : P(X) → P(X) on a set X such that [ [ ∅ ] ] = ∅ and, forall x ∈ X, [ [ {x} ] ] = {x}, the set of fixed points of [ [ • ] ] is a convexity C on X for such that [ [ • ] ] C = [ [ • ] ].
From here on, if no confusion is possible, the convex hull operator associated to a given convexity will mostly be written

[ [ • ]
] and we will write

[ [ p 1 , • • • , p m ] ] for [ [ {p 1 , • • • , p m } ] ]. Definition 2.1.4. A convex space (X, C) is recursive if, for all non empty finite subset S of X and for all point p 0 ∈ X [ [ S ∪ {p 0 } ] ] = p∈[ [ S ] ] [ [ p, p 0 ] ]. (4)
[ [ S ] ] • * p p 0 [ [ p0, p ] ] [ [ S ∪ {p 0 } ] ] (5)
Let us say that a convexity C on a set X is an interval convexity if, for all C ⊂ X, the following two conditions are equivalent :

(1)

C ∈ C (2) ∀p, q ∈ C [ [ p, q ] ] ⊂ C (6)
Lemma 2.1.5. A recursive convexity C on a set X is an interval convexity.

Proof. Let S = {p 0 , • • • , p m } be an arbitrary non empty subset of C. Assume that, for all p, q ∈ C, [ [ p, q ] ] ⊂ C. We show, by induction on the cardinality of S, that

[ [ S ] ] ⊂ C. If m ≤ 1 there is nothing to show. Assume that m > 1; from D = [ [ S \ {p m } ] ] ⊂ C and from [ [ S ] ] = ∪ q∈D [ [ p m , q ] ] we have [ [ S ] ] ⊂ C; by Lemma 2.1.3, C is convex; 2 Definition 2.1.6. A convex space (X, C) has Property C if, for all triple (a, b 1 , b 2 ) of points of X and all points c ∈ [ [ a, b 1 ] ] and d ∈ [ [ c, b 2 ] ] there exists a point b ∈ [ [ b 1 , b 2 ] ] such that d ∈ [ [ a, b ] ] • b a • d • •b 1 • b 2 • c [ [ a, b 1 ] ] [ [ a, b 2 ] ] (7)
] ] = b∈[ [ b1,b2 ] ] [ [ a, b ] ] and d ∈ [ [ c, b 2 ] ] ⊂ [ [ a, b 1 , b 2 ] ] there exists b ∈ [ [ b 1 , b 2 ] ] such that d ∈ [ [ a, b ] ].
Assume now that C is an interval convexity for which Property C holds and let us show that C is recursive.

Since S 0 ∪{p 0 } ⊂ p∈[ [ S0 ] ] [ [ p, p 0 ] ] it is enough to see that p∈[ [ S0 ] ] [ [ p, p 0 ] ] is convex since we then have [ [ S 0 ∪{p 0 } ] ] ⊂ [ [ p∈[ [ S0 ] ] [ [ p, p 0 ] ] ] ] = p∈[ [ S0 ] ] [ [ p, p 0 ] ]. Let p 1 , p 2 be two points of p∈[ [ S0 ] ] [ [ p, p 0 ] ]. We show that [ [ p 1 , p 2 ] ] ⊂ p∈[ [ S0 ] ] [ [ p, p 0 ] ].
There exist

q 1 , q 2 ∈ [ [ S 0 ] ] such that p i ∈ [ [ p 0 , q i ] ] ; let c ∈ [ [ p 1 , p 2 ] ].
Applying Property C to the configuration

         a = p 0 c 1 = p 1 b 1 = q 1 c 2 = b 2 = p 2 we find a point b ∈ [ [ q 1 , p 2 ] ] such that c ∈ [ [ p 0 , b ] ].
With the configuration

         a = p 0 c 1 = b 1 = q 1 b 2 = q 2 c 2 = p 2 we find a point b ∈ [ [ q 1 , q 2 ] ] such that b ∈ [ [ p 0 , b ] ]. From c ∈ [ [ p 0 , b ] ] and b ∈ [ [ p 0 , b ] ] we have c ∈ [ [ p 0 , b ] ]. From b ∈ [ [ q 1 , q 2 ] ] and q 1 , q 2 ∈ [ [ S 0 ] ] we have b ∈ [ [ S 0 ] ] and finally c ∈ p∈[ [ S0 ] ] [ [ p, p 0 ] ] and therefore [ [ p 1 , p 2 ] ] ⊂ p∈[ [ S0 ] ] [ [ p, p 0 ] ]. 2
Given a set X let X {0,1} be the family of non empty finite subsets of X of cardinality at most 2 ; given a map I : X {0,1} → P(X)\{∅} let General convex spaces are more akin to general topological spaces or measure spaces, in that the structure is defined by a family of subsets, than to the usual linear convex sets where the convex structure is associated to some previously defined algebraic structure. This of course does not preclude a given convexity structure from being associated to an underlying algebraic structure as, for example, it is the case for tropical, or linear, convexity. In any case, in the absence of an underlying primary structure from which the convexity structure is derived, it seems natural to define a morphism of convex spaces as it is done for topological spaces (continuous maps) or measure spaces (measurable maps) ; morphisms of general convex spaces are called, following Van Mill and Van de Vell, convexity preserving maps. A convexity preserving map from a convex space (X 1 , C 1 ) to a convex space (X 2 , C 2 ) is a map ϕ :

C I = C ⊂ X : ∀{x, y} ⊂ C I({x, y}) ⊂ C .
X 1 → X 2 such that, for all C ∈ C 2 , ϕ -1 C ∈ C 1 .
Obviously, the class of convex spaces and convexity preserving maps constitute a category, whose study is not the subject matter of this paper. All we will need is the associated notion of isomorphism : two convex spaces (X 1 , C 1 ) and (X 2 , C 2 ) are isomorphic if there exists a bijective convexity preserving map ϕ : X 1 → X 2 whose inverse is also convexity preserving; such a map is naturally called an isomorphism of convexity spaces. Clearly, Both Property C and the Pash-Peano Property are properties of polytopes spanned by at most three points; one could informally say that, for interval convexities, the Kakutani property is a "2-dimentional property". One can compare with Proposition 2.2.8 below. Definition 2.2.6. A convex space (X, C) has the Generalized Pash-Peano Property if for all non empty finite subsets B 1 and B 2 of X and all triple of points (a,

if ϕ : (X 1 , C 1 ) → (X 2 , C 2 ) is an isomorphism then so is ϕ -1 : (X 2 , C 2 ) → (X 1 , C 1 ) and ϕ : (X 1 , C 1 ) → (X 2 , C 2 ) is an isomorphism if and only if it is bijective and for all C 1 ∈ C 1 all C 2 ∈ C 2 , ϕ C 1 ∈ C 2 and ϕ -1 C 2 ∈ C 1 .

Algebraic separation of convex sets

2 , c 1 , c 2 ) of points of X such that c i ∈ [ [ a, b i ] ], one has [ [ b 1 , c 2 ] ] ∩ [ [ b 2 , c 1 ] ] = ∅. a • •b 1 • b 2 • c 1 • c 2 [ [ a, b 1 ] ] [ [ a, b 2 ] ] [ [ b 1 , c 2 ] ] [ [ b 2 , c 1 ] ] (8
c 1 , c 2 ) with c i ∈ [ [ {a}∪B i ] ] one has [ [ {c 1 } ∪ B 2 ] ] ∩ [ [ {c 2 } ∪ B 1 ] ] = ∅.
A proof of the following proposition can be found in [START_REF] Horvath | Toplogical convexities, selections and fixed points[END_REF].

Proposition 2.2.7. A convex space (X, C) has the Kakutani Property if and only if it has the Generalized Pash-Peano Property.

Given a convex space (X, C) and a convex set P ∈ C the family P ∩C : C ∈ C is a convexity on P called the induced convexity. Proposition 2.2.8 below is due to Keimel and Wieczorek; their proof can be found [START_REF] Keimel | Kakutani property of polytopes implies Kakutani property of the whole space[END_REF].

Proposition 2.2.8. A convex space (X, C) has the Kakutani Property if and only if all polytopes have, with respect to the induced convexity, the Kakutani Property.

Proof. Notice that the Generalized Pash-Peano Property holds if and only if it holds on polytopes. 2 Property C is due to Coppel, [START_REF] Coppel | Foundations of Convex Geometry[END_REF] where some of the results stated here, albeit in a somewhat different form and for a different purpose can also be found. Convexities with the Kakutani Property are studied in [START_REF] Coppel | Foundations of Convex Geometry[END_REF] and [START_REF] Van De | Theory of convex structures[END_REF]; Holmes in [START_REF] Holmes | Geometric functional analysis and its applications[END_REF] explicitely uses the Pash-Peano Property -without giving it a name since in a real vector space it is obvious -to prove the algebraic separation theorem.

Tropical convexity

A semiring is a set A, with two operations ⊕ : A × A → A and : A × A → A and two distinguished elements 1 A and 0 A such that: (1) (A, ⊕, 0 A ) is an abelian semigroup.

(2) for all x, y, z ∈ A, z (x y) = (z x) y.

(3) for all x, y, z ∈ A, z

(x ⊕ y) = (z x) ⊕ (z y). (4) for all x ∈ A, 0 A x = 0 A and 1 A x = x
For simplicity we will assume that is commutative.

An idempotent semiring, also called a tropical semiring or a Maslov semiring is a semiring A such that, for all x ∈ A, x ⊕ x = x. A (tropical) semifield is a (tropical) semiring for which (A , , 1 A ) is a group (A = A\{0 A }). Since tropical multiplication can sometimes be the "usual addition" or the "usual multiplication" of scalars we will denote, for an arbitrary semifield (A , , 1 A ), by u inv the inverse of u ∈ A .

An idempotent abelian semigroup (A, ⊕, 0 A ) is a semilattice with smallest element 0 A , and reciprocally, where x ⊕ y is the least upper bound of {x, y} with respect to the partial order x y if x ⊕ y = y.

Some tropical semirings (1)(maxplus semiring)

A = R ∪ {-∞}, 0 max+ = -∞, 1 max+ = 0, x ⊕ y = max{x, y}, x y = x + y. On could could also take A to be Z ∪ {-∞} or Q ∪ {-∞}. (2)(maxtimes semiring) A = R + , 0 max× = 0, 1 max× = 1, x ⊕ y = max{x, y}, x y = xy. One could also take for A N or Q + .
(3)(distributive lattice) L, ∨, ∧, 0 L , 1 L is a disributive lattice with a smallest and a largest element; let x ⊕ y = x ∨ y and x y = x ∧ y.

The maxplus and the maxtimes semirings are semifields ; if u ∈ R the multiplicative inverse of u in the maxplus semifield is -u ; if u ∈ R + \{0} = R ++ the multiplicative inverse of u in the maxtimes structure is u -1 .

A (tropical) semimodule over a (tropical) semiring (A, ⊕

A , A , 0 A , 1 A ) is an abelian (idempotent) semigroup M, ⊕, 0 M endowed with an operation : A × M → M such that : ∀a, b ∈ A ∀x, y ∈ M          a (x ⊕ y) = (a x) ⊕ (b y) (a ⊕ A b) x = (a x) ⊕ (b x) (a A b) x = a (b x) 1 A x = x and 0 A x = 0 M
>From here on we will mostly drop the subscripts on the operations ⊕, . Tropical semirings are also called idempotent semirings or Maslov semirings.

A tropical semiring A is a totally ordered tropical semiring if for all x, y ∈ A, x ⊕ y ∈ {x, y}.

The maxplus semiring A ∪ {-∞} and the maxtimes times semiring A + where A ∈ {Z, Q, R, • • • } are totally ordered.

Some tropical semimodules

(1) Let A be an arbitrary semiring and S a non empty set. On M = A S , the set of all functions from S to A, let ⊕ and be the pointwise operations: (f ⊕ g)(s) = f (s) ⊕ g(s) and (a f )(s) = a f (s) ; and take 0 M be the function which is identically 0 A . (1.2) If A is the maxtimes semiring R + and S is as above we have the maxtimes semimodule R + ) n .

(2) Let (L, ∨) be a real vector space semilattice, for example the space of real valued bounded functions on a given set X or the set of real continuous functions on a compact set X where ∨ = max; on L + = {x ∈ L : 0 x} let x y = x ∨ y and, for t ∈ R + , t x = tx. With these operations L + is a semimodule over the maxtimes semiring R + .

A subset C of a tropical semimodule M over a tropical semiring A is tropically convex if :

∀(x, y) ∈ C 2 ∀(a, b) ∈ A 2 such that a ⊕ b = 1 (a x)
S = {x 1 , • • • , x m } of M , [ [ S ] ] = m i=1 a i x i : (a 1 , • • • , a m ) ∈ A n s.t. m i=1 a i = 1 Proof. Let I({x, y}) = (a x) ⊕ (b y) : (a, b) ∈ A 2 and a ⊕ b = 1 . L 0 holds since, with a ⊕ b = 1, we have (a x) ⊕ (b x) = (a ⊕ b) x = 1 x = x.
The following straightforward calculation shows that (2) of Lemma 2.1.9 holds. Assume that

a i ⊕ b i = 1 and a ⊕ b = 1. Let w = (a 1 x) ⊕ (b 1 y) and z = (a 2 x) ⊕ (b 2 y) with a i ⊕ b i = 1. Then (a w) ⊕ (b z) = [(a a 1 ) ⊕ (b a 2 )] x ⊕ [(a b 1 ) ⊕ (b b 2 )] y . >From [(a a 1 ) ⊕ (b a 2 )] ⊕ [(a b 1 ) ⊕ (b b 2 )] = a (a 1 ⊕ b 1 ) ⊕ b (a 2 ⊕ b 2 ) = a 1 ⊕ b 1 = a ⊕ b = 1 we have (a w) ⊕ (b z) ∈ I({x, y}).
A similar straightforward calculation yields the second part. 2

Let C trop (M ) denote the family of tropically convex subsets of a tropical semimodule M over a tropical semiring A; the emptyset belongs to C trop (M ). Proposition 3.0.2. If M is a tropical semimodule over a tropical semifield A then tropical convexity on M is recursive.

Proof. . We have seen that C trop (M ) is an interval convexity ; let us see that Property C holds. Let c and d be two points of M such that

     (1) c = (s a) ⊕ (t b 1 ) with s ⊕ t = 1 and (2) d = (s c) ⊕ (t b 2 ) with s ⊕ t = 1 (9)
which gives The structure of halfspaces and separation theorems, algebraic and analytic, in maxplus convexity are studied in [START_REF] Briec | Separation in B-convexity[END_REF], [START_REF] Briec | Halfspaces and Hahn-Banach like properties in B-convexity and max-plus convexity[END_REF], [START_REF] Cohen | Hahn-Banach separation theorem for max-plus semimodules[END_REF], [START_REF] Cohen | Hahn-Banach separation theorem for max-plus semimodules[END_REF] and [START_REF] Nitica | The structure of max-plus hyperplanes[END_REF].

d = (s s) a ⊕ ((s t) b 1 ) ⊕ (t b 2 ) If (s t) ⊕ t = 0 let u = (s t) ⊕ t inv . Then [(s t) b 1 ⊕ (t b 2 )] = ((s t) ⊕ t ) (t 1 b 1 ) ⊕ (t 2 b 2 ) with t 1 = u (s t) and t 2 = u t . Furthermore, t 1 ⊕ t 2 = [u (s t)] ⊕ (u t ) = u [(s t) ⊕ t ] = 1. We have shown that (t 1 b 1 ) ⊕ (t 2 b 2 ) ∈ [ [ b 1 , b 2 ] ]. Let b = (t 1 b 1 ) ⊕ (t 2 b 2 ). From d = (s s) a ⊕ ((s t) ⊕ t ) b and (s s) ⊕ (s t) ⊕ t = s (s ⊕ t) ⊕ t = s ⊕ t = 1 we have d ∈ [ [ a
As is well known, maxplus convexity in R ∪ {-∞} n and B-convexity

(that is maxtimes convexity in R n + ) are isomorphic: ln(x 1 , • • • , x n ) = (ln(x 1 ), • • • , ln(x n )), with ln(0) = -∞, is a convexity preserving bijection from R n + to R ∪ {-∞} n whose inverse is E(x 1 , • • • , x n ) = (e x1 , • • • , e xn ), with 0 = e -∞
, is also convexity preserving.

Since R n and R n ++ are, respectively, convex subsets of the maxplus semimodule R ∪ {-∞} n and the maxtimes semimodule R n + and, from E R n = R n ++ , one has that maxplus convexity on R n and B-convexity on R n ++ are also isomorphic.

More tropical semimodules and tropical convex sets

Apart from the standard finite dimensional maxplus convexity on R ∪ {-∞} n or maxtimes convexity R n + there are natural (possibly) infinite dimensional tropical convexities, for example:

(1) The "Hom functor" yields tropical semimodule: given a two tropical semimodules M 1 and M 2 over a tropical semiring A one can consider the set Hom A (M 1 , M 2 ) of all maps ψ :

M 1 → M 2 such that (1) ∀(a, x) ∈ A × M 1 ψ(a x) = a ψ(x) (2) ∀(x 1 , x 2 ) ∈ M 1 × M 1 ψ(x 1 ⊕ x 2 ) = ψ(x 1 ) ⊕ ψ(x 2 )
There is on Hom A (M 1 , M 2 ) an obvious tropical A-semimodule structure :

(a ψ)(x) = a ψ(x) and (ψ 1 ⊕ ψ 2 )(x) = ψ 1 (x) ⊕ ψ 2 (x). For all ψ ∈ Hom A (M 1 , M 2 ), ψ(0) = 0; let ker ψ = {x ∈ M 1 : ψ(x) = 0}, it is a sub-semimodule of M 1 .
Elements of Hom A (M 1 , M 2 ) are convexity preserving maps (as previously defined) but also, the image

ψ(C) of an element C ∈ C trop (M 1 ) by a map ψ ∈ Hom A (M 1 , M 2 ) belongs to C trop (M 2 ). A map θ : C 1 → M 2 from C 1 ∈ C trop (M 1 ), C 1 = ∅, to M 2 is A-affine if, for all x 1 , x 2 ∈ C 1 and for all a 1 , a 2 ∈ A such that a 1 ⊕ a 2 = 1 θ (a 1 x 1 ) ⊕ (a 2 x 2 ) = a 1 θ(x 1 ) ⊕ a 2 θ(x 2 )
The set Aff A (C 1 , M 2 ) of A-affine maps from C 1 to M 2 is, with respect to the pointwise and ⊕ operations, an A-semimodule (recall that the operation on A is assumed to be commutative). The image of a convex subset of

C 1 by θ ∈ Aff A (C 1 , M 2 ) is convex and the inverse images of convex subset of M 2 is a convex subset of C 1 . (2) Let us say that a subset K of an A-semimodule M is a cone in M if: (1) ∀(x 1 , x 2 ) ∈ K × K x 1 ⊕ x 2 ∈ K (2) ∀(a, x) ∈ A × K a x ∈ K A cone in M is a convex subset of M (if a 1 ⊕ a 2 = 1 then either a 1 = 0 or a 2 = 0). If K is a cone in M then K ∪ {0}
is a subsemimodule of M and if A is a semifield and L is a sub-semimodule of M then L = L\{0} is a cone in M . For example, given a nonempty set X, let, for all a ∈ A, c a ∈ A X be the constant map c a (x) = a; A X , with the pointwise operations is an A-semimodule and if A is a semifield {c a : a ∈ A } is a cone in A X (which we identify with A ) and A X is also a cone in A X which contains A . Given two cones K 1 ⊂ M 1 and K 2 ⊂ M 2 let Hom A (K 1 , K 2 ) be the set of maps ψ :

K 1 → K 2 such that (1) ∀(x 1 , x 2 ) ∈ K 1 × K 1 ψ(x 1 ⊕ x 2 ) = ψ(x 1 ) ⊕ ψ(x 2 ) (2) ∀(a, x) ∈ A × K 1 ψ(a x) = a ψ(x) If A is a semifield then the set Hom A (K 1 , K 2 ) is itself a cone in the A-semimodule A X . If K is a subcone of A X such that c 1 ∈ K let M A (K) = ψ ∈ Hom A (K, A ) : ψ(c 1 ) = 1 Clearly, if ψ ∈ M A (K) then, for all a ∈ A , ψ(c a ) = a.
To each x ∈ X one can associate the map "evaluation at x", ψ → δ x (ψ) = ψ(x); clearly, δ x ∈ Hom A (K, A ) and, δ x (c 1 ) = 1 ; this shows that M A (K) is not empty. If ψ 1 and ψ 2 are in M A (K) then ψ 1 ⊕ ψ 2 is in M A (K) and, if A is a semifield then, for all a ∈ A and all ψ ∈ M A (K), a ψ is in M A (K). In other words, if A is a semifield then M A (K) is a cone in the A-semimodule A K , as a matter of facts a subcone of A K . If one takes A = R ∪ {-∞} and if X is a compact topological space then C(X), the space on real valued continuous functions on X, is a maxplus-cone in R ∪ {-∞} X then M A C(X) is the cone of Maslov's measures on X or idempotent measures on X , for its topological structure when X is a compact metric space see [START_REF] Bazylevych | Spaces of idempotent measures of compact metric spaces[END_REF].

a C = {a x : x ∈ C} C 1 ⊕ C 2 = {u ⊕ v : (u, v) ∈ C 1 × C 2 }. For a ∈ A and C 1 , C 2 ∈ C • trop (M ), C 1 ⊕ C 2 ∈ C • trop (M ) and a (C 1 ⊕ C 2 ) = a C 1 ⊕ a C 2 and, for a, b ∈ A and C ∈ C • trop (M ),      a C ∈ C • trop (M ) (a b) C = a (b C) 1 C = C and 0 C = {0} That (C 1 ⊕ C 2 ) ⊕ C 3 = C 1 ⊕ (C 2 ⊕ C 3 ) is also clear as well as the inclusion (a ⊕ b) C ⊂ (a C) ⊕ (b C). If A is a semifield then (a C) ⊕ (b C) ⊂ (a ⊕ b) C. Indeed if z = a x ⊕ b y with x, y ∈ C either a ⊕ b = 0 or a ⊕ b = 0. In the first case a = b = 0 and (a ⊕ b) C = 0 C = {0} while (a C) ⊕ (b C) = {0} ⊕ {0} = {0}. If a ⊕ b = 0 let c = (a ⊕ b) inv ; then (c a) x ⊕ (c b) y ∈ C and z = (a ⊕ b) (c a) x ⊕ (c b) y ∈ (a ⊕ b) C. In conclusion: if A is a semifield then C • trop (M ),
2 of M let [ [ S 1 | S 2 ] ] = [ [ T 1 ⊕ T 2 : T 1 × T 2 ⊂ S 1 × S 2 T 1 × T 2 = ∅} ] ] . We have [ [ S 1 | S 2 ] ] ∈ Poly • trop (M ) and [ [ S 1 | S 2 ] ] ⊂ [ [ S 1 ] ] ⊕ [ [ S 2 ]
]. Let us show by induction on the cardinality of

S 1 that [ [ S 1 ] ] ⊕ [ [ S 2 ] ] ⊂ [ [ S 1 | S 2 ] ]. Let S 1 = {x 1 , • • • , x k } and S 2 = {y 1 , • • • , y m }, u = ⊕ k i=1 a i x i , v = ⊕ m j=1 b j y j and ⊕ k i=1 a i = ⊕ m j=1 b j = 1. If k = 1 then u ⊕ v = x 1 ⊕ (⊕ m j=1 b j y j ) = ⊕ m j=1 b j (x 1 ⊕ y j ) ∈ [ [ S 1 | S 2 ]
]. Let k = n + 1. By hypothesis A is totally ordered and ⊕ n+1 i=1 a i = 1, one of the a i is therefore equal to 1; without loss of generality one can assume that a 1 = 1. Let u 1 = ⊕ n i=1 a i x i ; from ⊕ n i=1 a i = 1 and the induction hypothesis we have

u 1 ⊕ v ∈ [ [ S 1 \{x n+1 } | S 2 ] ] ⊂ [ [ S 1 | S 2 ] ]. Let us write u 1 ⊕ v = ⊕ l j=1 d j T 1j ⊕ T 2j
where ⊕ l j=1 d j = 1 and, for all j, T 1j is a nonempty subset of S 1 \{x n+1 } and T 2j is a nonempty subset of S 2 .

Then

u 1 ⊕ v ⊕ x n+1 = ⊕ l j=1 d j [T 1j ⊕ x n+1 ] ⊕ T 2j and therefore u 1 ⊕ v ⊕ x n+1 ∈ [ [ S 1 | S 2 ] ]. >From 1 ⊕ a m+1 = 1 we have (u 1 ⊕ v) ⊕ [a m+1 (u 1 ⊕ v)] = u 1 ⊕ v and from the convexity of [ [ S 1 | S 2 ] ] we have (u 1 ⊕ v) ⊕ a m+1 (u 1 ⊕ v ⊕ x n+1 ) ∈ [ [ S 1 | S 2 ] ]. But (u 1 ⊕ v) ⊕ a m+1 (u 1 ⊕ v ⊕ x n+1 ) = (u 1 ⊕ v) ⊕ a m+1 (u 1 ⊕ v) ⊕ a m+1 x m+1 = (u 1 ⊕ v) ⊕ a m+1 x m+1 = u ⊕ v.
For the topological structure of the space of nonempty compact maxplus convex subsets of a power of the real line see [START_REF] Bazylevych | Hyperspaces of max-plus convex subsets of power of the real line[END_REF] and the references cited therein.

Convexity and topology

Given a set X a convexity C and a topology τ what kind of meaningful relationship can one impose between C and τ ? Connectedness of polytopes seems natural but, in itself, might not be sufficient to yield the fixed point property or other properties related to convexity in normed or locally convex topological vector spaces. Nonetheless, we will recall at the end of this section that connectedness is still good enough to capture some of the fundamental results of classical nonlinear analysis.

The standard n dimensional simplex, that is

{(t 0 , • • • , t n ) ∈ [0, 1] n+1 : n i=0 t i = 1} will be denoted, as usual, by ∆ n : the boundary ∂∆ n of ∆ n is {(t 0 , • • • , t n ) ∈ ∆ n : n i=0 t i = 0}. The family of nonempty subsets of {0, • • • , n} is denoted by n and, for J ∈ n , ∆ n,J = {(t 0 , • • • , t n ) ∈ ∆ n : ∀i ∈ J t i = 0} and ∆ -j n = {(t 0 , • • • , t n ) ∈ ∆ n : t j = 0}.
Let us recall that a topological space Z is homothopically trivial if, for all n ∈ N and all continuous map g : ∂∆ n → Z there exists a continuous map G : ∆ n → Z whose restriction to ∂∆ n is g. A topological space Z is contractible if there exists a continuous map h : [0, 1] × Z → Z such that h(0, -) : Z → Z is constant and h(1, -) : Z → Z is the identity map; contractible spaces are homothopically trivial.

A convex topological space is a triple (X, τ, C) where τ is a topology on X and C is a convexity on X. Two convex topological spaces (X 1 , τ 1 , C 1 ) and (X 2 , τ 2 , C 2 ) are homeomorphic convex topological spaces if there exists an homeomorphism ϕ : (X 1 , τ 1 ) → (X 2 , τ 2 ) which is also an isomorphism of the convex spaces (X 1 , C 1 ) and (X 2 , C 2 ). For example, x → ln(x) is an homeomorphism from the B-convex space R n ++ to the maxplus convex space R n ; one could extend this homeomorphism from R n + to R ∪ {-∞} n by taking as a neighbourhood base for -∞ in R ∪ {-∞} the family of complements of closed half rays [r, +∞[, r ∈ R; which makes R ∪ {-∞} n homeomorphic to R + .

It is not hard to see that B-polytopes in R n + are compact and contractible, see [START_REF] Briec | Nash points, Ky Fan inequality and equilibria of abstract economies in Max-Plus and B-convexity[END_REF], and obviously metrizable ; the same is therefore true of maxplus-polytopes in R n .

A family of topological results has played a fundamental role in nonlinear analysis and mathematical economics; these results pertain mainly to the existence of a fixed point (for a single valued or a multivalued map), to the existence of a continuous selection (or an approximate continuous selection) for a multivalued map or to the nonvacuity of the intersection of a given family of sets. Most of these results can be extended to topological convex spaces whose polytopes are homothopically trivial. One of these results is as easily stated as it is fundamental, in its classical version it can be seen as an avatar of Brouwer's fixed point theorem: it is the Knaster-Kuratowski-Mazurkiewicz theorem, also known as the KKM Lemma, whose standard form can be found in [START_REF] Dugundji | Fixed Point Theory[END_REF], [START_REF] Aubin | Optima and Equilibria, An Introduction to Nonlinear Analaysis[END_REF], [START_REF] Aubin | L'analyse non linéaire et ses motivations économiques[END_REF], [START_REF] Border | Fixed points theorems with applications to economics and game theory[END_REF] or [START_REF] Singh | Fixed point theory and best approximations: the KKM-map principle[END_REF] along with its various forms and applications. The classical "KKM Lemma" reads as follows:

( KKM) If {F 0 , • • • , F n } is a family of closed sets of ∆ n such that, for all J ∈ n , ∆ nJ ⊂ ∪ j∈J F j then ∩ n j=0 F j = ∅.
Lassonde noticed that the KKM Lemma holds if, instead of being closed, all the F i are open [START_REF] Lassonde | Sur le principe KKM[END_REF]; but see also [START_REF] Fan | Some properties of convex sets related to fixed point theorems[END_REF] and [START_REF] Shih | Covering properties of convex sets[END_REF]. We will still refer to the KKM Lemma for either a closed or an open covering as the "classical KKM".

Some basic intersection theorems in topological convex spaces

Given a topological convex space X, τ, C let us say that a subset

A
, • • • , x n } of X, not necessarily distincts, such that, for all nonempty subset J ⊂ {0, • • • , n}, [ [ {x j : j ∈ J} ] ] ⊂ ∪ j∈J F j . Proof. Let P = [ [ {x 0 , • • • , x n } ]
] and, for all J ∈ n , let

P J = [ [ {x i : i ∈ J} ] ].
As in [START_REF] Horvath | Contractibility and Generalized Convexity[END_REF], Theorem 1, one shows that there exists a continuous map θ :

∆ n → P such that, ∀J ∈ n , θ ∆ n,J ⊂ [ [ {x i : i ∈ J} ] ]. Let M j = θ -1 (P ∩ F j ) ; from P ⊂ ∪ n j=0 F j we have ∆ n = ∪ n j=0 M j
, • • • , x n } of X, not necessarily distincts, such that, [ [ {x 0 , • • • , x n } ] ] ⊂ ∪ n j=0 F j and, for all j ∈ {0, • • • , n}, [ [ {x i : i = j} ] ] ⊂ F j then ∩ n i=0 F i = ∅ . Proof. Let P -j = [ [ {x i : i = j} ]
]; all the other symbols retain the meaning they had in the previous proposition. Since P ⊂ ∪ n j=0 F j we have ∆ n = ∪ n j=0 M j and we also have, for all j ∈ {0,

C i = ∅, then ∩ n i=0 C i = ∅. Proof. For each i ∈ {0, • • • , n} pick a point x i ∈ ∩ i =j C j ; we then have [ [ {x 0 , • • • , x n } ] ] ⊂ ∪ n j=0 C j and, for all j ∈ {0, • • • , n}, {x i : i = j} ⊂ C j . From the Alexandrov-Pasynkov Theorem, ∩ n i=0 M i = ∅ and consequently, ∩ n i=0 C i = ∅. 2
Klee's Theorem [START_REF] Klee | On certain intersections of convex sets[END_REF] for the usual linear convexity was proved independently by Berge in 1959, [START_REF] Berge | Espaces topologiques et fonctions multivoques[END_REF] or [START_REF] Berge | Topological spaces[END_REF], using the Hahn-Banach Theorem.

Let us say that a subset S of a convex topological space X, C is starshaped if there exists a point x 0 ∈ S such that, for all x ∈ S, [ [ x 0 , x ] ] ⊂ S; in which case we say that S is starshaped at x 0 . A convex set is starshaped. The next result is a topological version of Breen's Theorem [START_REF] Breen | Starshaped unions and nonempty intersections of convex sets in R d[END_REF]. Proof. For each J ∈ n let F J = ∪ j∈J U j ; choose x J ∈ F J such that F J is starshaped at x J . Let P = x J : J ∈ n } and S J = P ∩ F J ; notice that S J is starshaped at

x J . Given J 1 , • • • , J l ∈ n let J = J 1 ∪ • • • ∪ J l and notice that S J1 ∪ • • • ∪ S J1 = S J .
The family S J : J ∈ n is made of starshaped subsets of the polytopes P and either all of its members are closed in P or all of its members are open in P ; furthermore, by hypothesis, all of the sets S J are homothopically trivial subsets of P . Enumerate n arbitrarily, n = J 0 , • • • , J m }; since, for all {J i1 , • • • , J i k }, ∪ k j=1 S Ji j = S J l with J l = ∪ k j=1 J ij we see that ∪ k j=1 S Ji j is a starshaped subset of P . For all λ ∈ {0, • • • , m}, let T λ = S J λ = ∪ j∈J λ S j ; T λ is an homothopically trivial subset of P and the family

T λ : λ ∈ {0, • • • , m} is union closed. For all Λ ∈ m let G Λ = ∪ λ∈Λ T λ . If Λ ⊂ Λ then G Λ ⊂ G Λ
and these sets are homothopically trivial subsets of P . There exists a continuous map θ : ∆ m → P such that, for all Λ ∈ m , θ ∆ m,Λ ⊂ G Λ . Let M λ = θ -1 (T λ ); notice that ∆ m,Λ ⊂ ∪ λ∈Λ M λ and that, either all the sets M λ are closed in ∆ m or all of them are open in ∆ m . By the standard KKM Theorem, ∩ m λ=0 M λ = ∅ and therefore ∩ m λ=0 T λ = ∅ that is ∩ J∈ n S J = ∅ and a fortiori,

∩ J∈ n F J = ∅ from which ∩ n i=0 U i = ∅. 2
Breen's original result is on the one hand more precise than Proposition 4.1.4 and on the other hand less general. She considered a family F of nonempty compact starshaped sets in R d and proved that ∩F = ∅ if the union of every subfamily of F of d + 1 or fewer members of F has starshaped union. On the other hand, Proposition 4.1.4 holds in arbitrary topological vector spaces. For a topological proof of Breen's original theorem see [START_REF] Horvath | Intersection of sets with n-connected unions[END_REF].

One can show that tropical starshaped sets in R d , that is maxplus starshaped sets, are homothopically trivial, as well as B-convex (maxtimes) starshaped sets in R d + . We leave aside for now the question of a completely analoguous "tropical Breen's Theorem" in R d or in R d + .

There is another Proposition whose proof is somewhat similar to that of Proposition 4.1.4; to keep this section at a reasonable length the proof will not be given here.

Fixed points theorems

The results of this section, given without proofs, rely on the fact that maxplus polytopes in R n and Bpolytopes in R n + are contractible, and therefore homothopically trivial.

The first result is a tropical version of the Fan-Browder Theorem, page 143 in [START_REF] Dugundji | Fixed Point Theory[END_REF].

An arbitrary product

n i=1 [a i , b i ] of real intervals is a B-convex set in R n + consequently, an arbitrary nonempty B-convex set in R n
+ has a neighbourhood base consisting of B-convex sets ; B-convex set are obviously metrizable. By the corollary on page 259 of [START_REF] Horvath | Extension and selection theorems in topological spaces with a generalized convexity structure[END_REF], arbitrary nonempty B-convex set in R n + are absolute retracts. The same is true of arbitrary nonempty maxplus convex sets in R n . Since we are here in finite dimention we could also reach the same conclusion from the fact that B-convex sets are contractible and locally contractible.

Back to the begining

The two previous sections lead naturally to the following question: when does a convex topological space have homothopically trivial polytopes ? As we have already observed, maxplus polytopes in R n , or B-polytopes in R n + are homothopically trivial, as well as usual linear polytopes, and all of this can easily be shown by hand without appealing to any general theory. Let us nonetheless reformulate our question as follows: what kind of "compatibility " properties should link the convexity and the topology of a given convex topological space X, τ, C in order to have homothopically trivial polytopes ? Of course, we would like the standard tropical convexities, maxplus in R n or maxtimes in R n + , to have those properties. Two answers will be given, one geometric, in terms of the existence of a compatible geodesic structure, and one topological. For simplicity, we will restrict our attention to metric spaces eventhough it is not necessary, and even not recommanded, think of convex sets in locally convex topological vector spaces. In everything that follows (X, d) is a metric space and C is a convexity on X.

Uniform geodesic convexities

Let us say that (X, d, C) is a uniform geodesic space if C is an interval convexity and if there exists a continuous function λ : X × X × [0, 1] → X such that ∀(x 0 , x 1 , t, s)

      
(1) λ(x 0 , x 1 , 0) = x 0 and λ(x 0 , x 1 , 1) = x 1 (2) d λ(x 0 , x 1 , t), λ(x 0 , x 1 , s) = |t -s|d(x 0 , x 1 )

(3) λ x 0 , x 1 , [0, 1] = [ [ x 0 , x 1 ] ]
If (X, d, C) is a uniform geodesic space then all nonempty convex subsets of X are, with respect to the induced convexity and the induced metric, uniform geodesic spaces. These are indeed very strong conditions, and stronger than they need to be, but as they stand they hold for maxplus convexity in R n and B-convexity in R n ++ . In the first case, the metric is given d(x, y) = x -x ∨ y ∞ + y -x ∨ y ∞ , one can check that it is the so called Hilbert affine distance (Stephane Gaubert showed to the author that maxplus segments in R n are geodesics for the Hilbert affine distance) that is, for x, y ∈ R n , d(x, y) = max i {0, x i -y i } + max i {0, y i -x i }. Conditions (1) and ( 3) imply together that starshaped sets are homothopically trivial, as a matter of fact they are contractible. In a uniform geodesic space one can naturally define the midpoint function µ : X × X → X by µ(x 0 , x 1 ) = λ(x 0 , x 1 , 1/2). From (2) we have d(x 0 , µ(x 0 , x 1 )) = d(x 1 , µ(x 0 , x 1 )) = 1/2d(x 0 , x 1 ).

2. 1 .

 1 The general frameworkDefinition 2.1.1. A convexity on a set X is a family C of subsets of X such that:(Conv1) ∅ ∈ C and X ∈ C and, ∀x ∈ X, {x} ∈ C; (Conv2) if A is a subfamily of C then A belongs to C; (Conv3) if A is an updirected 1 subfamily of C then A belongs to C.Elements of C are called convex sets and (X, C) is a convex space. Given A ⊂ X let [ [ A ] ] C = {C ∈ C : A ⊂ C} ; It is the C-hull of A. By Conv1 and Conv2, ∀A ⊂ X [ [ A ] ] C ∈ C and [

Definition 2 . 2 . 1 .

 221 A convex space (X, C) has the Pash-Peano Property if, for all quintuple (a, b 1 , b

( 1 . 1 )

 11 If A is the maxplus semiring R ∪ {-∞} and if S = {1, • • • , n} we have the maxplus semimodule R ∪ {-∞} n ;

  ⊕ (b y) ∈ C Proposition 3.0.1. The set of convex sets of a given semimodule M over an arbitrary tropical semiring A is an interval convexity on M for which, for all x, y ∈ M , [ [ x, y ] ] = (a x) ⊕ (b y) : (a, b) ∈ A 2 and a ⊕ b = 1 . Furthermore, for an arbitrary finite and non empty subset

7 Proposition 3 . 0 . 4 .

 7304 , b ] ]. Proposition 3.0.4 and Corollary 3.0.5 below follow from the previous propositions of this section and propositions 2.2.4 and 2.2.If M is a tropical semimodule over a totally ordered tropical semifield A then tropical convexity on M has the Kakutani Property and the Generalized Pash-Peano Property. Corollary 3.0.5. An arbitrary convex subset of a tropical semimodule over a totally ordered tropical semifield is the intersection of all the half spaces containing it.

  Let us denote by C • trop (M ) the set of nonempty convex sets of a tropical semimodule M over a tropical (commutative) semiring A; define on C • trop (M ) the operations and ⊕ as follows:

2 Proposition 4 . 1 . 2 (

 2412 and also ∆ n,J ⊂ ∪ j∈J M j and finally, from the classical KKM Lemma, ∩ n j=0 M j = ∅. Alexandrov-Pasynkov). In a topological convex space X, τ, C with homothopically trivial polytopes let {F 0 , • • • , F n } be a family of subsets of X all of which are finitely closed or all of which are finitely open. If there exists a family of points {x 0

Proposition 4 . 1 . 4 (

 414 Breen's Theorem). Let X, τ, C be a convex topological space in which starshaped subsets of polytopes are homothopically trivial. Let {U 0 , • • • , U n } be a family of subsets of X all of which are finitely closed or all of which are finitely open. If, for all nonempty subsets J ⊂ {0, • • • , n} the set ∪ j∈J U j is starshaped then ∩ n j=0 U j = ∅.

Proposition 4 . 1 . 5 .

 415 Let X, τ, C be a convex topological space in which starshaped subsets of polytopes are homothopically trivial. Let {U 0 , • • • , U n } be a family of subsets of X all of which are finitely closed or all of which are finitely open. If, for all nonempty subsets J ⊂ {0, • • • , n} of cardinality at most n the set ∩ j∈J U j is starshaped and if ∪ n j=0 U j is starshaped then ∩ n j=0 U j = ∅.

Proposition 4 . 2 . 1 .

 421 Let C be either a compact maxplus convex set in R n or a compact B-convex set in R n + and let T : C → C be a multivalued map with closed nonempty maxplus (respectively, B) convex values. If, for all y ∈ C, the set T -1 y = {x ∈ C : y ∈ T x} is open in C then there exists x ∈ C such that x ∈ T x. Proof. It is a consequence of Proposition 4.1.1; see [19] Theorem 3 for the details. 2

Proposition 4 . 2 . 2 .Proposition 4 . 2 . 3 .

 422423 Let C be either a maxplus convex set in R n or a B-convex set in R n + and f : C → C a continuous map whose image is contained in a compact subset of C. Then f has a fixed point.Proof. From the Generalized Schauder Theorem, page 165 in[START_REF] Dugundji | Fixed Point Theory[END_REF]. 2We close this section with Kakutani's Fixed Theorem in maxplus or B-convexity. First, in R n + open balls with respect to the • ∞ norm are B-convex sets and, for all B-convex subset C of R n + and all r > 0 the r-neighbourhood of C with respect to • ∞ are B-convex; in the language of[START_REF] Horvath | Extension and selection theorems in topological spaces with a generalized convexity structure[END_REF], B-convex sets are l.c. spaces. The same is true of maxplus convex sets in R n . Let C be either a compact maxplus convex set in R n or a compact B-convex set in R n + and T : C → C an an upper semicontinuous multivalued map with nonempty closed maxplus (respectively, B) convex values. Then, there exists x ∈ C such that x ∈ T x.Proof. Page 262 of[START_REF] Horvath | Extension and selection theorems in topological spaces with a generalized convexity structure[END_REF]. 2

Proposition 4 . 3 . 1 .

 431 In a uniform geodesic space (X, d, C) a closed set C ⊂ X is convex if and only if it contains the midpoint of any two of its points, that is µ C × C) ⊂ C. Proof. The nontrivial part of the proof is done by induction on the set of dyadic numbers in the real interval [0, 1]. 2 One can see without too much difficulty that, for all maxplus convex segments [ [ x, y ] ] ⊂ R n , [ [ x, y ] ] = [ [ x, x⊕y ] ]∪ [ [ x⊕y, y ] ] and therefore, a subset C of R n is maxplus convex if and only if (1) ∀x, y ∈ C x ⊕ y ∈ C (C is a subsemilattice of R n ) (2) ∀x, y ∈ C x ≤ y ⇒ [ [ x, y ] ] ⊂ C.

Lemma 2.1.9. Assume that condition L 0 holds

  For all {x, y} ⊂ X and for all {w, z} ⊂ I({x, y}), I({w, z}) ⊂ I({x, y}).

	The proofs of Lemmas 2.1.8 and 2.1.9 below are left to the reader.
	Lemma 2.1.8. X, C I is a convex space if and only if the following condition holds:
	L 0 : ∀x ∈ X I({x}) = {x}
	in which case it is an interval convexity.
	The convex interval [ [ x, y ] ] C I , which is the intersection of all the C I convex sets containing {x, y}, does not
	have to be I({x, y}).
	. The following statements are equivalent:
	(1) For all {x, y} ⊂ X [ [ x, y ] ] C I = I({x, y}).
	(2)

) Definition 2.2.2. A

half-space of a convex space (X, C) is a convex set H ∈ C whose complement X \H is also in C.

Definition 2.2.3. A

  convex set (X, C) has the Kakutani Property if, for all pair (C 1 , C 2 ) of disjoint non empty convex sets there exists a half-space H such that C 1 ⊂ H and C 2 ⊂ X \H. A recursive convex space (X, C) has the Kakutani Property if and only if it has the Pash-Peano Property.

		X	•
		•
	H 1		C 2
	C 1	H 2
	Kakutani Property
	Theorem 2.2.4.

Proof. A proof can be found in

[START_REF] Horvath | Toplogical convexities, selections and fixed points[END_REF]

. 2 Corollary 2.2.5. If an interval convexity C on a set X has Property C and the Pash-Peano Property then it has the Kakutani Property. Proof. By Proposition 2.1.7. 2

  What might not be at first so clear is that P 1 ⊕ P 2 is a polytope if P 1 and P 2 are polytopes. We will show that this is the case if A is a totally ordered semifield and therefore, if A is a totally ordered semifield then Poly • trop (M ) is a tropical semimodule over A as a matter of fact, a tropical subsemimodule of C • trop (M ). The set T ⊂ M being finite and nonempty let us write T for x∈T x. Given two nonempty finite subsets S 1 and S

endowed with elementwise multiplication by elements of A and elementwise addition is a tropical semimodule over A.

(5) Let Poly • trop (M ) be the set of nonempty tropical polytopes of the A-semimodule M ; it is a subset of

C • trop (M ) to which {0} belongs. If P = [ [ S ]

] where S is finite nonempty subset of M then, for all a ∈ A, a P = [ [ {a x : x ∈ S} ] ] an therefore a P is a polytope.

  of X is finitely closed (resp. open) if, for all polytope P ∈ C, A ∩ P is closed (resp. open) in P with respect to the induced topology. (KKM). In a topological convex space X, τ, C with homothopically trivial polytopes let {F 0 , • • • , F n } be a family of subsets of X all of which are finitely closed or all of which are finitely open. Then ∩ n i=0 F i = ∅ if and only if there exists a family of points {x 0

	Proposition 4.1.1

  In a topological convex space X, τ, C with homothopically trivial polytopes let {C 0 , • • • , C n } be a family of convex sets of X of which are finitely closed or all of which are finitely open. If, ∪ n i=0 C i is convex and, for all j ∈ {0, • • • , n}, ∩ i =j

	• • • , n}, ∆ -j n ⊂ M j and finally, from
	the classical Alexandrov-Pasynkov Theorem, ∩ n j=0 M j = ∅. 2
	Proposition 4.1.3 (Klee's Theorem).

If (s t) ⊕ t = 0 then (s t) = t = 0. Since A is a semifield, either s = 0 or t = 0 ; from s ⊕ t = 1 and t = 0 we have s = 1 and therefore t = 0 and, from s ⊕ t = 1, s = Proof. Let c i = (s i a) ⊕ (t i b i ) with s i ⊕ t i = 1. We have to show that there exists (u 1 , v 1 ) and (u 2 , v 2 ) in

Substituting for c 1 and c 2 , respectively, in

To complete the proof it is sufficient to find (u i , v i ) such that

Assume that either s 1 = 0 or s 2 = 0; let us say s 2 = 0.

Given an arbitrary u

Equations (a), (b) and (c) hold. We have to see that u 1 can be chosen such that

Since A is totally ordered and s i ⊕ t i = 1 there are only four cases to consider:

From this remark and Proposition 4.3.1 one gets the following caracterization of closed maxplus convex subsets of R n .

where (x, y) → µ max (x, y) is the midpoint function on R n associated to the Hilbert affine metric.

One can show that, if x y then µ max (x, y) i = max x i , y i -max 1≤j≤n {y j -x j } 2 .

Uniform convexities

We restrict the discussion to metric spaces, but uniform topological spaces would do as well. Let C be a convexity on a metric space (X, d) such that polytopes are compact. The convexity is completely specified by the convex hull operator restricted to the set of nonempty finite subsets of X; let X be that space and let K(X) be the space of nonempty compact subsets of X. Both X and K(X) are metric spaces with respect to the Hausdorff metric associated to the metric d on X. Let us say that X, d, C) is a uniform convex space if the convex hull operator {x 0 ,

] is uniformly continuous from X to K(X). One can show that B-convexity on R n + is a uniform convexity; the same is true of maxplus convexity on R n . Proposition 4.3.3. Let X, d, C) be a convex space with compact polytopes. If X, d, C) is a uniform convex space, if polytopes are connected and if the Kakutani Separation Property holds then convex sets are homothopically trivial.

The proof is somewhat too involved to be presented here; it can be found in [START_REF] Horvath | Toplogical convexities, selections and fixed points[END_REF], for uniform topological spaces, or in [START_REF] Van De | A selection theorem for topological convex structures[END_REF] for metric spaces. All of the classical selection, extension and fixed point theorems for locally convex and normed spaces, as well as some topological properties, like being an absolute retract, hold in the context of uniform convexities. It has been said at the begining of this discussion that a convexity has the Kakutani Separation Property if and only if arbitrary polytopes have it; therefore, all the results alluded to follow from properties of polytopes ("finite dimensional properties") and one global property: uniform continuity of the convex hull operator.