Andrea Tommasi

Mickael Knoll

Alain Vauchez

J Signorelli

Catherine Thoraval

Roland E Logé

Structural reactivation in

Structural reactivation in plate tectonics controlled by olivine crystal anisotropy

Andréa Tommasi1 * , Mickael Knoll 1,2 , Alain Vauchez 1 , Javier W. Signorelli 3 , Catherine Thoraval 1 and Roland Logé [START_REF] Vauchez | Why do continents break up parallel to ancient orogenic belts?[END_REF] Reactivation of structures inherited from previous collisional or rifting events, especially lithospheric-scale faults, is a major feature of plate tectonics. Its expression ranges from continental break-up along ancient collisional belts [START_REF] Wilson | Did the Atlantic close and then re-open?[END_REF][START_REF] Vauchez | Why do continents break up parallel to ancient orogenic belts?[END_REF] to linear arrays of intraplate magmatism and seismicity [START_REF] Sykes | Intraplate seismicity, reactivation of preexisting zones of weakness, alkaline magmatism, and other tectonism postdating continental fragmentation[END_REF][START_REF] Zoback | Recurrent intraplate tectonism in the New Madrid Seismic Zone[END_REF] . Here we use multiscale numerical models to show that this reactivation can result from an anisotropic mechanical behaviour of the lithospheric mantle due to an inherited preferred orientation of olivine crystals. We explicitly consider an evolving anisotropic viscosity controlled by the orientation of olivine crystals in the mantle. We find that strain is localized in domains where shear stresses on the inherited mantle fabric are high, and that this leads to shearing parallel to the inherited fabric. During rifting, structural reactivation induced by anisotropy results in oblique extension, followed by either normal extension or failure. Our results suggest that anisotropic viscosity in the lithospheric mantle controls the location and orientation of intraplate deformation zones that may evolve into new plate boundaries, and causes long-lived lithospheric-scale wrench faults, contributing to the toroidal component of plate motions on Earth.

Texture-induced viscoplastic anisotropy is an intrinsic feature of crystalline materials deforming by dislocation creep. It entails a directional dependence of the mechanical strength on the crystal orientation. This behaviour, expressed macroscopically as an anisotropic viscosity, results from both the anisotropy of the elastic tensor and the discrete nature of dislocation glide along densely packed lattice directions on selected crystallographic planes. It depends on the crystal symmetry and, at the aggregate scale, on the orientation of the constitutive crystals or texture. As they have fewer slip systems, low-symmetry minerals are more anisotropic than cubic metals. Orthorhombic olivine is highly anisotropic: it has only three independent slip systems. At high temperature, an olivine crystal deforms up to 100 times faster if it activates the 'easy' (010)[100] system rather than the 'hard' (010)[001] system [START_REF] Durham | Plastic flow of oriented single crystals of olivine. 1. Mechanical data[END_REF][START_REF] Bai | High-temperature creep of olivine single crystals. 1. Mechanical results for buffered samples[END_REF] . This leads to development of strong crystal preferred orientations (texture), which are ubiquitous in naturally and experimentally deformed mantle rocks [START_REF] Tommasi | Viscoplastic self-consistent and equilibrium-based modelling of olivine lattice preferred orientations. Implications for upper mantle seismic anisotropy[END_REF] . Moreover, seismic anisotropy provides evidence for coherent orientation of olivine crystals at the scale of hundreds of kilometres in the upper mantle [START_REF] Wuestefeld | Shear-wave splitting database[END_REF] . Yet, in contrast to the large number of studies on the role of texture-induced plastic anisotropy on the deformation of cubic and hexagonal metallic alloys since the early work of ref. 9, plastic anisotropy has been largely ignored in earth sciences.

Models considering a transversely isotropic viscosity showed nevertheless that anisotropy changes the convection length scales, leads to strain localization, and modifies the initiation times of convective instabilities in the mantle [START_REF] Christensen | Some geodynamical effects of anisotropic viscosity[END_REF][START_REF] Honda | Strong anisotropic flow in a finely layered asthenosphere[END_REF][START_REF] Lev | Rayleigh-Taylor instabilities with anisotropic lithospheric viscosity[END_REF] . Mechanical anisotropy due to olivine crystal preferred orientations in the mantle was also proposed to explain the reactivation of ancient collisional structures during continental rifting [START_REF] Vauchez | Why do continents break up parallel to ancient orogenic belts?[END_REF] . Models of the deformation of a homogeneous, anisotropic mantle subjected to an axisymmetric tension show a strong directional softening, leading to strain localization and development of shearing when extensional stresses are oblique to the pre-existing mantle fabric [START_REF] Tommasi | Continental rifting parallel to ancient collisional belts: An effect of the mechanical anisotropy of the lithospheric mantle[END_REF] . These models avoided simplifications to the viscosity tensor by using a viscoplastic self-consistent (VPSC) formalism [START_REF] Lebensohn | A self-consistent anisotropic approach for the simulation of plastic deformation and texture development of polycrystals: Application to zirconion alloys[END_REF] to relate the crystal and largescale plastic anisotropies, but they did not allow for the evolution of olivine textures and, hence, of the anisotropy in response to the new solicitation.

To fully account for an evolving anisotropic viscosity as a function of the orientation of olivine crystals in the mantle, we coupled the VPSC formalism to FORGE2005, a commercialized three-dimensional finite-element mechanical (FEM) code. The coupling is performed through a set of 1,000 olivine crystals associated with each finite element. After each FEM time increment, the local velocity gradient tensor is used as a boundary condition for the VPSC simulation, leading to evolution of the polycrystal texture and viscosity. The updated polycrystal viscosity tensor is then used to calculate the stress field in the next FEM time step (see Supplementary Fig. S1).

To analyse the deformation of a plate containing a pre-existing lithospheric-scale transcurrent shear zone or transpressional belt, we run a series of models using a plate with aspect ratio 1.2×1×0.1 (Fig. 1) subjected to a constant divergent velocity or tensional stress on its left lateral boundary. Free-slip conditions (normal velocities and shear stresses are null) are imposed on the opposite boundary. Boundaries normal to the y and z axes are free (normal and tangential stresses are null). We use a multidomain meshing tool to define a planar zone, 0.1 or 0.2 units wide, cross-cutting the entire plate, at 45 • (Fig. 1) or normal to the x axis (see Supplementary Fig. S2). This zone has an initial 'wrench-fault-type' olivine texture [START_REF] Tommasi | Upper mantle tectonics: Three-dimensional deformation, olivine crystallographic fabrics and seismic properties[END_REF] ; maximum concentrations of [100] and [010] axes are horizontal, respectively parallel and normal to the shear-zone trend (Fig. 1). This texture is coherent with shear-wave splitting data in large-scale strike-slip faults and transform boundaries, where fast S waves are polarized parallel to the trend of the fault up to 50 km away from the surface expression of the fault [START_REF] Vauchez | Intraplate Strike-Slip Deformation Belts[END_REF] , and in collisional belts [START_REF] Wuestefeld | Shear-wave splitting database[END_REF][START_REF] Tommasi | Upper mantle tectonics: Three-dimensional deformation, olivine crystallographic fabrics and seismic properties[END_REF][START_REF] Vauchez | Mountain building: Strike-parallel displacements and mantle anisotropy[END_REF] . For numerical stability reasons, the initial texture is however significantly weaker (J index = 3, Fig. 1, where J index is a measure of the texture intensity, expressed as the volume-averaged integral of the squared orientation densities) than eq ε .

x'y' ε .

y'y' ε . zz ε ε ε ε External domain ε . eq ε . xy ε . xx ε . zz ε ε ε ε External domain ε . eq ε . xy ε . xx ε . zz ε ε ε ε 90° shear zone ε .
eq ε .

x'y' ε .

y'y' ε . zz ε ε ε ε
Figure 2 | Evolution of strain rates in the inherited texture zone (thick lines) and in the surroundings (thin lines) as a function of the imposed extension. a, Inherited 'shear zone' at 45 • the imposed extension. b Inherited 'shear zone' normal to the imposed extension. Analysis of the Von Mises equivalent strain rate εeq = √ 2/3ε ij εij highlights the evolution of the strain distribution in response to the reorientation of the olivine crystal preferred orientations in both domains. The latter also results in changes in the deformation regime, outlined by the evolution of the rates of shearing parallel to the inherited shear zone (ε x y ), extension normal to it (ε y y ) and vertical thinning (ε zz ).

those from naturally deformed peridotites [START_REF] Tommasi | Viscoplastic self-consistent and equilibrium-based modelling of olivine lattice preferred orientations. Implications for upper mantle seismic anisotropy[END_REF] (2 < J index < 26, peak at 8). Olivine texture in the surrounding medium is initially random.

All models show an anisotropic behaviour, characterized by dependence of strength on the direction of solicitation relative to the texture orientation. Strain localization arises owing to lateral variations in texture orientation and intensity. Equivalent strain rates (ε eq ) are higher when the texture orientation relative to the imposed extension results in high resolved shear stresses on the 'easy' (010)[100] and (001)[100] slip systems in most crystals within a domain. Strain is therefore localized in the 'inherited texture domain' at 45 • to the imposed extension (Figs 1 and2a), whereas a similar domain normal to the extension has a higher initial strength, deforming more slowly than the surrounding medium (Fig. 2b).

Decomposition of the horizontal flow field into its poloidal (divergence) and toroidal (vorticity) components highlights that viscoplastic anisotropy produces lateral variations in the poloidal flow field (strain localization) and toroidal flow (strike-slip deformation) in the 'inherited texture domain' (Fig. 3). The latter deforms therefore by transtension (Fig. 2): shearing parallel to its trend (ε x y ) accompanies stretching normal to it (ε y y ) and vertical thinning (ε zz ). The intensity of the toroidal component depends to the first order on the orientation of the texture relative to the stress field, but also on the texture intensity. It is higher in 'inherited texture domains' oblique to the extension direction and for stronger initial textures (see Supplementary Fig. S3). Analysis of the results of the model with an 'inherited texture domain' normal to the viscoplastic anisotropy due to preferred orientation of olivine crystals triggers the reactivation of pre-existing lithospheric-scale faults or transpressional belts if their orientation enables higher shear stresses on the olivine crystals' easy slip systems within them than in the surroundings. Strain localization in the present models is limited: the strain-rate contrast between the domain with inherited texture and the initially isotropic surroundings is less than 2 (Fig. 2). This contrast is enhanced if the inherited texture is stronger, like those usually observed in naturally deformed peridotites (see Supplementary Fig. S3), or if the surroundings have a non-random initial texture in a 'hard' orientation relatively to the solicitation, but texture-induced mechanical anisotropy remains less effective in producing strain localization than changes in deformation mechanism due to grain refinement [START_REF] Braun | A simple parameterization of strain localization in the ductile regime[END_REF] , viscous heating [START_REF] Regenauer-Lieb | Positive feedback of interacting ductile faults from coupling of equation of state, rheology and thermal-mechanics[END_REF] or damage processes [START_REF] Bercovici | Tectonic plate generation and two-phase damage: Void growth versus grain size reduction[END_REF] . However, the latter are observed neither in hightemperature, high-pressure experiments nor in nature and the former processes require an already localized forcing, that is, a pre-existing localization of the deformation [START_REF] Montési | A unified description of localization for application to large-scale tectonics[END_REF] , usually produced in geodynamical models by the introduction of ad hoc temperature anomalies or 'weak domains'. Viscoplastic anisotropy due to coherent orientation of olivine crystals, in contrast, is an intrinsic feature of lithospheric plates as shown by shear-wave splitting [START_REF] Wuestefeld | Shear-wave splitting database[END_REF] . We propose that lateral variations in mechanical behaviour due to changes in the inherited olivine crystal preferred orientations within a plate represent a major 'seed' process for strain localization extension shows however that even a slight departure from perfect symmetry of the texture (see Supplementary Fig. S2) produces toroidal flow (Fig. 3) and hence horizontal shearing within the inherited texture domain (Fig. 2b).

Evolution of olivine textures with increasing strain modifies the strain distribution. Progressive rotation of olivine [100] and [010] axes towards the maximum finite extension and shortening directions, respectively (Fig. 1), results in geometrical hardening in both the 'inherited texture domain' and the surroundings (Fig. 2), because shear stresses on the 'weak' (010)[100] and (001)[100] slip systems decrease. Faster texture evolution in the 'inherited texture domain' in the 45 • model (the surroundings in the 90 • one) results in rehomogenization of the strain distribution (inversion of the strength contrast) at high strain. Texture evolution and hence the mechanical behaviour depend on the strain regime: extension or compression normal to the dominant orientation of the easy slip systems produce hardening, whereas shearing parallel to it produces softening (see Supplementary Fig. S4). Reorientation of the olivine texture also changes the deformation regime in the 'inherited texture domain' (Fig. 2). In the 45 • model, the shear rate parallel to the trend of the 'inherited texture domain' (ε ˙x y ) decreases and the extension rate normal to it (ε ˙y y ) increases; the deformation evolves from transtensional to extensional (Fig. 2a).

Although extremely simple (they do not consider thermal effects and test only end-member geometries), these models show that creating the conditions for the activation of additional efficient localizing processes. In contrast to lateral variations of the geotherm, crustal thickness, or damage zones associated with brittle faults that may also seed localization, its lifetime is not limited by heat diffusion, isostatic re-equilibration, or erosion. Coherence between crustal structures of all ages and seismic anisotropy data suggest that texture-induced anisotropy may be preserved for more than 1 Gyr. It may therefore explain the reactivation of lithosphericscale strike-slip or transpressional zones hundreds of Myr old, as observed in the South Atlantic and East African rifting [START_REF] Vauchez | Why do continents break up parallel to ancient orogenic belts?[END_REF][START_REF] Tommasi | Continental rifting parallel to ancient collisional belts: An effect of the mechanical anisotropy of the lithospheric mantle[END_REF] .

Deformation controlled by texture-induced viscoplastic anisotropy associated with lithospheric-scale wrench faults and transpressional belts has a characteristic signature: development of shearing parallel to the trend of the reactivated structures. Rift initiation controlled by anisotropy-induced reactivation of shear zones results in transtension followed by normal extension. This evolution is observed in the East African rift [START_REF] Bosworth | Stress field changes in the Afro-Arabian rift system during the Miocene to Recent period[END_REF] , the North Atlantic margins [START_REF] Geoffroy | Tectonic evolution of the Greenland-Scotland Ridge during the Paleogene; new constraints[END_REF] , the Rhine graben [START_REF] Schumacher | Upper Rhine Graben: Role of preexisting structures during rift evolution[END_REF] and the eastern Gondwana break-up [START_REF] Powell | Pre-breakup continental extension in East Gondwanaland and the early opening of the eastern Indian Ocean[END_REF] . It supports reactivation of neoproterozoic belts during the eastern Gondwana and East African rifting, of Caledonian structures in the North Atlantic opening and of Variscan wrench faults in the Rhine graben. It also suggests that the role of mechanical anisotropy decreases in mature rift systems, in agreement with the present models' prediction that evolution of the olivine texture leads to progressive geometric hardening and, hence, to rift failure, if thermo-mechanical or magmatic processes leading to further softening are not activated in the thinned lithospheric domain.

As it favours shearing parallel to the pre-existing mantle structure, viscoplastic anisotropy may also explain the reactivation of large-scale wrench faults during successive collisional and extensional episodes, such as the Newfoundland-Azores-Gibraltar fault zone, which acted as a dextral strike-slip boundary during the Hercynian orogeny [START_REF] Keppie | Terranes in the Circum-Atlantic Paleozoic Orogens[END_REF] and was reactivated as a major transform accommodating the differential motion between Africa and Europe during the Central Atlantic opening. The large proportion of strike-slip focal mechanisms in intraplate seismic arrays, such as the New Madrid seismic zone in the southeastern US (ref. 27) or the present seismic activity along the Hercynian South Armorican shear zone in France [START_REF] Mazabraud | Evidence for short-scale stress field variations within intraplate central-western France[END_REF] , also suggests that the reactivation of the NE-SW-and NW-SE-trending mantle fabric imaged by shear-wave splitting in the southeastern US (ref. 29) and Brittany [START_REF] Judenherc | The Hercynian collision in the Armorican Massif; evidence of different lithospheric domains inferred from seismic tomography and anisotropy[END_REF] probably has a significant role in this intraplate deformation process.

A major consequence of viscoplastic anisotropy is that the stress and strain tensor eigenvectors are parallel only when the olivine texture is perfectly symmetric relatively to the stress field. Even a slight departure from symmetry produces shearing parallel to the average orientation of the dominant olivine slip system. Reactivation of a strike-slip or transpressional mantle texture therefore produces toroidal flow. The toroidal-poloidal partitioning depends on the orientation of the texture relative to the solicitation. In the present models (Fig. 3), the ratio between the vorticity and the divergence of the horizontal velocity field ranges from 0.2 to 0.3 (0.7 for a stronger initial olivine texture, Supplementary Fig. S3). These values encompass the average toroidal-poloidal partitioning ratio in the Earth (0.3-0.4; ref. 31), suggesting that texture-induced viscoplastic anisotropy in the mantle may significantly contribute to the Earth mantle toroidal flow and to belt-parallel transport recorded by both strain partitioning in the crust and fast-shear-wave polarization in past and present convergent boundaries [START_REF] Vauchez | Mountain building: Strike-parallel displacements and mantle anisotropy[END_REF] .

Methods

A VPSC formalism [START_REF] Lebensohn | A self-consistent anisotropic approach for the simulation of plastic deformation and texture development of polycrystals: Application to zirconion alloys[END_REF] is used to relate the crystal and polycrystal mechanical behaviours and to predict the texture evolution. Interaction between the crystal and the polycrystal is calculated using the Eshelby formalism; each crystal is considered as an inclusion embedded in a homogeneous equivalent medium that behaves as the average of all crystals. At the crystal scale, deformation is accommodated by dislocation glide on discrete slip systems, whose relative strength-or critical resolved shear stress-depends on the temperature, pressure and chemical environment. In the present simulations, critical resolved shear stresses (τ 0 ) and stress exponents (n) derived from high-temperature experiments on dry olivine crystals [START_REF] Durham | Plastic flow of oriented single crystals of olivine. 1. Mechanical data[END_REF][START_REF] Bai | High-temperature creep of olivine single crystals. 1. Mechanical results for buffered samples[END_REF] were used: τ , are added to ensure convergence. They never accommodate more than 1% of the total strain. Three-dimensional FEM modelling. FORGE2005 is a commercialized FEM software using an updated Lagrangian scheme. The finite-element formulation is based on a mixed velocity-pressure formulation with an enhanced (P1 + /P1) four-node tetrahedral element. Thermal effects are not considered. The mesh used in the present simulations is composed of about 12,000 linear tetrahedral elements. To define regions showing different initial textures, we use a multidomain mesh tool based on a level-set function [START_REF] Bernacki | Level set framework for the numerical modelling of primary recrystallization in polycrystalline materials[END_REF] . Heterogeneous mesh refinement allows a fine description of the interfaces with a minimum number of mesh elements. Resolution tests using about 9,000, 12,000, and 25,000 linear elements corroborate that strain localization does not depend on the mesh; the strain rate is roughly constant within each domain (small variations are observed close to the free boundaries) and the ratio between the average equivalent strain rates in the inherited texture domain and in the surroundings are similar. Low-resolution tests with about 4,500 elements show in contrast significantly weaker strain localization. To enable comparison between models, strain rates are normalized by the initial Von Mises equivalent strain rate, εeq = √ 2/3ε ij εij , of the surrounding medium, which has an initially random texture and hence an isotropic behaviour.

Figure 1 |

 1 Figure 1 | Geometry, boundary conditions and shear-strain ε xy distribution in a model with an 'inherited shear zone' at 45 • of the imposed extension direction after a total stretching of 40%. Texture-induced anisotropy results in strain localization and higher shear strain in the 'inherited shear zone'. Evolution of the olivine texture within the 'inherited shear zone' may be evaluated by comparing the initial (top left) and final (bottom right) textures. Bottom left: slip systems in the olivine crystal, 'easy' (010)[100] and (001)[100] systems are outlined by dark-grey shading and thicker lines parallel to the slip direction.

Figure 3 |

 3 Figure 3 | Poloidal-toroidal partitioning in the models. Maps of the divergence (a,c) and vorticity (b,d) of the horizontal flow field. Higher divergence values indicate strain localization in the 'inherited texture domain' or in the surroundings, when the former is at 45 • (a) or normal (b) to the imposed extension. Larger divergence variations denote stronger localization in the 45 • model. Toroidal flow, indicated by a non-null vorticity, is restricted to the 'inherited texture domain'. Divergence and vorticity variations are directly related to changes in the olivine texture, being roughly homogeneous within each domain.

= 1 /

 1 3τ [001](100) 0 and n = 3 for all systems. Two additional slip systems, which are not observed in olivine, { 111} 110 and {111} 110 , with τ 0 = 50 * τ [100](010) 0

Received 4

 4 February 2009; accepted 24 April 2009; published online 24 May 2009

Geosciences Montpellier, CNRS & Université de Montpellier

2, Pl. E Bataillon, 34095 Montpellier cedex 5, France, 2 MINES ParisTech, CEMEF-Center for Materials Forming, UMR CNRS 7635, BP207, 06904 Sophia Antipolis Cedex, France,[START_REF] Sykes | Intraplate seismicity, reactivation of preexisting zones of weakness, alkaline magmatism, and other tectonism postdating continental fragmentation[END_REF] Instituto de Fisica de Rosario, CONICET, Universidad Nacional de Rosario, 2000 Rosario, Argentina. *e-mail: andrea.tommasi@gm.univ-montp2.fr.

Acknowledgements

This study was partially funded by the programme Action Marges of the Institut National des Sciences de l'Univers, Centre National de la Recherche Scientifique (INSU-CNRS), France. Collaboration with J.W.S. was supported by a CNRS-CONICET cooperation program. M.K. benefited from a PhD scholarship from the Ministère de la Recherche et de l'Enseignement Supérieur, France.

Author contributions

This work is the outcome of a study on the effect of olivine fabrics on the mechanical behaviour of the continental lithosphere started by A.V. and A.T. M.K. ran all simulations as part of his Ph.D. under the supervision of A.T. and R.L. J.W.S. performed the coupling between the VPSC and the FEM codes. C.T. assisted in the analysis of the modelled flow fields.

Additional information

Supplementary information accompanies this paper on www.nature.com/naturegeoscience. Reprints and permissions information is available online at http://npg.nature.com/ reprintsandpermissions. Correspondence and requests for materials should be addressed to A.T.


Ilias Ftouhi

On the Cheeger inequality for convex sets

Keywords: Cheeger constant, complete systems of inequalities, Blaschke-Santaló diagrams, convex sets, sharp spectral inequalities AMS classification:: 52A10, 52A40, 49R99, 65K15

 in the case of thin triangles.

A celebrated inequality due to Jeff Cheeger states that for every open bounded set Ω ∈ R n (where n ≥ 2) one has:

λ 1 (Ω) ≥ 1 4 h(Ω) 2 ,
where λ 1 (Ω) is the first Dirichlet eigenvalue and h(Ω) is the Cheeger constant of Ω, which is defined as follows:

h(Ω) := inf P (E) |E| E measurable and E ⊂ Ω , (1)
where P (E) is the perimeter of De-Giorgi of E measured with respect to R n (see for example [START_REF] Parini | An introduction to the Cheeger problem[END_REF] for definitions) and |E| is the n-dimensional Lebesgue measure of E. Any set C Ω ⊂ Ω for which the infimum is attained is called (when it exists) a Cheeger set of Ω. We refer to [START_REF] Parini | An introduction to the Cheeger problem[END_REF] for an introduction to the Cheeger problem.

In the present paper, d and r respectively correspond to the diameter and the inradius functionals.

Recently, E. Parini [START_REF] Parini | Reverse Cheeger inequality for planar convex sets[END_REF] remarked that the constant 1 4 can be improved for the class K 2 (for every n ∈ N * , we denote K n the class of bounded convex bodies of R n of non-empty interior). He proved the following inequality:

∀Ω ∈ K 2 , λ 1 (Ω) ≥ π 2 16 h(Ω) 2 , (2)
and noted that the constant π 2 16 is also not optimal. He then took a shape optimization point of view by introducing the functional J : Ω ∈ K 2 -→ J(Ω) := λ1(Ω) h(Ω) 2 for which he proves the existence of a minimizer in K 2 and conjectures that it is the square; in which case the optimal lower bound would be given by:

min Ω∈K 2 J(Ω) = J (0, 1) 2 = 2π 2 (2 + √ π) 2 ≈ 1.387...
Nevertheless, as far as we know, as mentioned in [START_REF] Parini | Reverse Cheeger inequality for planar convex sets[END_REF]Section 6], the existence of an optimal shape in higher dimensions (n ≥ 3) remains open.

In order to obtain a lower bound of J on the class K n , one can combine the inequality h(Ω) ≤ n r(Ω) (which is obtained by taking the inscribed ball B r(Ω) as a test set in the definition of the Cheeger constant h(Ω)) and Protter's inequality [START_REF] Protter | A lower bound for the fundamental frequency of a convex region[END_REF]:

∀Ω ∈ K n , λ 1 (Ω) ≥ π 2 4 1 r(Ω) 2 + n -1 d(Ω) 2 , (3)
which generalises Hersch's inequality [START_REF] Hersch | Sur la fréquence fondamentale d'une membrane vibrante: évaluations par défaut et principe de maximum[END_REF] (used by Parini for the planar case) to higher dimensions. We then obtain the following lower bound:

∀n ≥ 2, ∀Ω ∈ K n , J(Ω) := λ 1 (Ω) h(Ω) 2 > π 2 4n 2 ,
which improves the original constant 1 4 given by J. Cheeger only for n ∈ {2, 3}. In which cases, we have:

∀Ω ∈ K 2 , J(Ω) > π 2 16
≈ 0.616... and ∀Ω ∈ K 3 , J(Ω) > π 2 36 ≈ 0.274...

In the present paper, we improve the Cheeger-Parini's inequality [START_REF] Blaschke | Konvexe Bereiche gegebener konstanter Breite und kleinsten Inhalts[END_REF]. Our result in this direction is stated as follows:

Theorem 1.1. We have:

∀Ω ∈ K 2 , J(Ω) = λ 1 (Ω) h(Ω) 2 ≥ πj 01 2j 01 + π 2 ≈ 0.902...
where j 01 denotes the first zero of the first Bessel function.

At last, we are interested by the question of the existence of an minimizer of J for higher dimensions n ≥ 3. We prove the following Theorem: Theorem 1.2. Let us define the real sequence (β n ) n as follows: ∀n ∈ N * , β n := inf Ω∈K n J(Ω).

We have:

1. (β n ) n is a decreasing sequence.

lim

n→+∞

β n = 1 4 .
3. For n ≥ 2, if the strict inequality β n < β n-1 holds, we have the following existence result:

∃Ω * n ∈ K n , J(Ω * n ) = inf Ω∈K n J(Ω).
Let us give a few interesting comments on Theorem 1.2:

• The convergence result lim n→+∞ β n = 1 4 of shows that the constant 1 4 given in the original Cheeger inequality [START_REF] Cheeger | A lower bound for the smallest eigenvalue of the Laplacian[END_REF] is optimal in the sense that there exists no constant C > 1 4 such that:

∀n ≥ 1, ∀Ω ∈ K n , λ 1 (Ω) h(Ω) 2 ≥ C.
• We believe that the assertion β n < β n-1 is true for any n ≥ 2. This conjecture is motivated by the discussion of Section 4.2. In particular, when n = 2, we have:

inf Ω∈K 2 J(Ω) < π 2 4 = inf ω∈K 1 J 1 (ω)
. Thus, we retrieve Parini's result of existence in the class of planar sets without using the explicit formulae of Cheeger constants of planar convex sets.

At last, we note that the result of Theorem 1.1 relies on the combination of Protter's inequality (3) and the Faber-Krahn inequality [START_REF] Faber | dass unter allen homogenen membranen von gleicher fl ache undgleicher spannung die kreisf örmige den tiefsten grundton gibt[END_REF][START_REF] Krahn | Über eine von rayleigh formulierte minimaleigenschaft des kreises[END_REF] to bound λ 1 (Ω) from below and an upper sharp estimate of the Cheeger constant in terms of the inradius and the area.

The study of complete systems of inequalities relating some given functionals is an interesting subject for its own. It is closely related to the so called Blaschke-Santaló diagrams, we refer to the original works of Blaschke [START_REF] Blaschke | Konvexe Bereiche gegebener konstanter Breite und kleinsten Inhalts[END_REF] and Santaló [START_REF] Santaló | On complete systems of inequalities between elements of a plane convex figure[END_REF] and to the more recent works [START_REF] Böröczky | Optimizing area and perimeter of convex sets for fixed circumradius and inradius[END_REF][START_REF] Delyon | The missing (A, d, r) diagram[END_REF][START_REF] Hernández Cifre | Is there a planar convex set with given width, diameter, and inradius?[END_REF][START_REF] Hernández Cifre | Optimizing the perimeter and the area of convex sets with fixed diameter and circumradius[END_REF][START_REF] Hernández Cifre | Some optimization problems for planar convex figures[END_REF] for some interesting examples involving geometrical functionals and to [START_REF] Ftouhi | Complete systems of inequalities relating the perimeter, the area and the Cheeger constant of planar domains[END_REF][START_REF] Ftouhi | Blaschke-santaló diagram for volume, perimeter, and first dirichlet eigenvalue[END_REF][START_REF] Lucardesi | On blaschke-santaló diagrams for the torsional rigidity and the first dirichlet eigenvalue[END_REF][START_REF] Van Den Berg | On capacity and torsional rigidity[END_REF][START_REF] Briani | Some inequalities involving perimeter and torsional rigidity[END_REF][START_REF] Van Den Berg | On the relations between principal eigenvalue and torsional rigidity[END_REF] for recent examples dealing with diagrams involving spectral and geometrical quantities.

In the present paper we provide a complete system of inequalities relating the Cheeger constant h, the inradius r and the area | • | of planar convex sets, which corresponds to a complete description of the related Blaschke-Santaló diagram introduced in Theorem 1.3.

Before stating the result, let us provide some notations and define various important notions: we denote d H the Hausdorff distance (for more details we refer for example to [START_REF] Henrot | Shape variation and optimization[END_REF]Chapter 2]), S n-1 ⊂ R n the unit sphere and B ⊂ R n a ball of unit volume. If Ω ∈ K n , we denote by h Ω : u ∈ S n-1 -→ sup In what follows we denote such a supporting hyperplane by H(Ω, u). We then have the following characterization of the supporting hyperplanes of Ω:

H(Ω, u) = {x ∈ R n , x, u = h Ω (u)}.
A point x ∈ ∂Ω is called regular if the supporting hyperplane at x is uniquely defined, that is if there is a unique u ∈ S n-1 such that x ∈ H(Ω, u) ∩ Ω. The set of all regular points of ∂Ω is denoted by reg(Ω). We also introduce U(Ω) the set of all outward pointing unit normals to ∂Ω at points of reg(Ω). We are now in position to define the form body Ω * of Ω as in [START_REF] Schneider | Convex Bodies: The Brunn-Minkowski Theory[END_REF]:

Ω * := u∈U (Ω) {x ∈ R n , x, u = 1}.
We are now ready to state the following result:

Theorem 1.3. We have:

∀Ω ∈ K 2 , 1 r(Ω) + πr(Ω) |Ω| ≤ h(Ω) ≤ 1 r(Ω) + π |Ω| , (4)
These inequalities are sharp as equalities are obtained for stadiums in the lower estimate and for domains that are homothetic to their form bodies in the upper one. Moreover, we have the following explicit description of the Blaschke-Santaló diagram:

D := 1 r(Ω) , h(Ω) | Ω ∈ K 2 and |Ω| = 1 = (x, y) x ≥ 1 r(B) = √ π and x + π x ≤ y ≤ x + √ π ,
where B ⊂ R 2 is a ball of unit area.

The present paper is organized as follows: in Section 2, we provide the proof of the sharp estimates of the Cheeger constant given in Theorem 1.3. Section 3 is devoted to the improvement of the Cheeger-Parini's inequality for planar convex sets (2), we also give improved results for some special shapes (triangles, rhombii and stadiums), see Proposition 3.1. We then prove the existence result of Theorem 1.2 in Section 4. We finally discuss some new sharp inequalities involving the first Dirichlet eigenvalue, the Cheeger constant, the inradius and the area of planar convex sets in Appendix 5. 2 Sharp estimates for the Cheeger constant: Proof of Theorem 1.3

The proof of Theorem 1.3 is presented in 3 parts:

The lower bound:

Let Ω ∈ K 2 , we denote C Ω ∈ K 2 its (unique) Cheeger set. Let us show that:

r(Ω) = r(C Ω ).
By the characterization of the Cheeger set of planar convex sets of [START_REF] Kawohl | Characterization of Cheeger sets for convex subsets of the plane[END_REF], we have

C Ω = Ω -1 h(Ω) + 1 h(Ω) B 1
, where B 1 is the ball of unit radius centred at the origin. We then have:

r(C Ω ) = r Ω -1 h(Ω) + 1 h(Ω) B 1 = r Ω -1 h(Ω) + r 1 h(Ω) B 1 = r(Ω) - 1 h(Ω) + 1 h(Ω) r(B 1 ) = r(Ω).
Since, the Cheeger set C Ω is convex, we can use the following Bonnesen's inequality [START_REF] Bonnesen | Theory of convex bodies[END_REF]:

P (C Ω ) ≥ πr(C Ω ) + |C Ω | r(C Ω ) ,
with equality if and only if C Ω is a stadium (note that does not mean that Ω is a stadium). Thus:

h(Ω) = P (C Ω ) |C Ω | ≥ πr(C Ω ) + |CΩ| r(CΩ) |C Ω | = πr(Ω) |C Ω | + 1 r(Ω) ≥ πr(Ω) |Ω| + 1 r(Ω) ,
where the last inequality is a consequence of the inclusion C Ω ⊂ Ω and thus is an equality if and only if Ω = C Ω . Finally, we proved the lower bound and the equality holds if and only if Ω is a stadium.

The upper bound:

Let Ω ∈ K 2 . We have by [START_REF] Matheron | La formule de Steiner pour les érosions[END_REF]Theorem 2]:

∀t ∈ 0, r(Ω) , |Ω -t | ≥ |Ω| 1 - t r(Ω) 2 , (5)
with equality if and only if Ω is homothetic to its form body.

If Ω is homothetic to its form body, we have by solving the equation

|Ω -t | = |Ω| 1 -t r(Ω) 2 = πt 2 : h(Ω) = 1 r(Ω) + π |Ω| .
From now on, we assume that Ω is not homothetic to its form body. Let us introduce the functions:

• f : t ∈ 0, r(Ω) -→ |Ω| 1 -t r(Ω) 2 -πt 2 = |Ω| -2|Ω| r(Ω) t + |Ω| r(Ω) 2 -π t 2 , • g : t ∈ 0, r(Ω) -→ |Ω -t | -πt 2 .
By (5), we have:

g(0) = f (0), ∀t ∈ 0, r(Ω) , g(t) > f (t).
This implies that 1/h(Ω), the first zero of g on [0, r(Ω)], is strictly larger than the first zero of f given by

1 r(Ω) + π |Ω| -1
(see Figure 2) , which proves the inequality.

The diagram:

The inequalities (4) imply that

D := |Ω| r(Ω) , |Ω|h(Ω) | Ω ∈ K 2 and |Ω| = 1 ⊂ (x, y) x ≥ 1 r(B) = √ π and x + π x ≤ y ≤ x + √ π .
It remains to prove the reverse inclusion. The proof follows the following steps:

1. We explicit a continuous family (S r ) r≤r(B) of convex bodies which fill the lower boundary of the diagram.

2. We explicit a continuous family (U r ) r≤r(B) of convex bodies which fill the upper boundary of the diagram.

3. We use the latter domains to construct (via Minkowski sums) a family of continuous paths (Γ r ) r≤r(B) which connect the upper domains to the lower ones and prove that we are able to cover all the area between the upper and lower boundaries.

Step 1: The lower boundary of the diagram:

As proved above, the lower boundary corresponds to stadiums. Let us consider the family of stadiums (Q t ) t≥0 given by convex hulls of the balls of unit radius centred in O(0, 0) and O t (t, 0). The functions

• t ∈ [0, +∞) -→ |Q t |/r(Q t ) = π + 2t • t ∈ [0, +∞) -→ |Q t |h(Q t ) = 2(π+t) √ π+2t .
are continuous and strictly increasing to infinity (when t → +∞). Thus, we have by the intermediate values Theorem:

|Q t | r(Q t ) , |Q t |h(Q t ) t ≥ 0 = (x, y) x ≥ 1 r(B) = √ π and y = x + π x .
Step 2: The upper boundary of the diagram:

Since equality h(Ω) = 1 r(Ω) + π
|Ω| holds for sets that are homothetical to their form bodies, we use such domains to fill the upper boundary.

Let us consider the family (C d ) d≥2 of the so-called symmetrical cup-bodies, which are given by convex hulls of the unit ball ( centred in O(0, 0) of radius 1) and the points of coordinates (-d/2, 0) and (d/2, 0). By using formula [START_REF] Brasco | Spectral inequalities in quantitative form[END_REF] of [START_REF] Hernández Cifre | Some optimization problems for planar convex figures[END_REF], we have for every d ≥ 2:

|C d | = d 2 -4 + π -2 arccos 2 d , r(C d ) = 1 and h(C d ) = 1 r(C d ) + π |C d | .
The functions

• d ∈ [2, +∞) -→ |C d |/r(C d ) = √ d 2 -4 + π -2 arccos 2 d , • d ∈ [2, +∞) -→ |C d |h(C d ) = √ d 2 -4 + π -2 arccos 2 d + √ π.
are continuous and strictly increasing to infinity (when d → +∞), this shows by the intermediate values Theorem that:

|C d | r(C d ) , |C d |h(C d ) d ≥ 2 = (x, y) x ≥ 1 r(B) = √ π and y = x + √ π .
Step 3: Continuous paths:

For every d ≥ 2 and t ≥ 0, we denote

C d := C d √ |C d | and Q t := Qt √ |Qt| the rescaled versions of C d and Q t (such that |C d | = 1 and |Q t | = 1). Since the functions • t ∈ [0, +∞) -→ 1 r(Q t ) = √ |Qt| r(Qt) = 2(π+t) √ π+2t • d ∈ [2, +∞) -→ 1 r(C d ) = √ |C d | r(C d ) = 2 √ d 2 -1 + 2 arcsin 2 d
are continuous and strictly increasing, we have that for every r ≤ r(B), there exists a unique

(t r , d r ) ∈ [0, +∞) × [2, +∞) such that r(Q tr ) = r(C dr ) = r.
From now on we denote S r := Q tp and L r := C dp .

For every r ≤ r(B), we introduce the closed and continuous path Γ r (see Figure 3) :

Γ r : [0, 3] -→ R 2 t -→        1/r(K t r ), h(K t r ) if t ∈ [0, 1], x r,1 (t), f 1 (x r,1 (t)) if t ∈ [1, 2], x r,2 (t), f 2 (x r,1 (t)) if t ∈ [2, 3],
where

• K t r := tSr+(1-t)Lr √ |tSr+(1-t)Lr| ∈ K 2 1 := {Ω ∈ K 2 , |Ω| = 1}, • x r,1 : t -→ (t -1) 1 r(B) + (2 -t) 1 r , • x r,2 : t -→ (3 -t) 1 r(B) + (t -2) 1 r , • f 1 : x ∈ [1/r(B), +∞) -→ x + √ π, • f 2 : x ∈ [1/r(B), +∞) -→ x + π x . The application t ∈ [0, 1] -→ tS r + (1 -t)L r ∈ (K 2 , d H
) is continuous and since the measure is continuous for the Hausdorff distance, we deduce that t ∈

[0, 1] -→ K t r ∈ (K 2 1 , d H ) is continuous,
]), the path t ∈ [0, 3] -→ Γ r (t) ∈ R 2 is a continuous curve.
We also notice that thanks to the linearity of the inradius for the Minkowski sum, as well as the Brunn-Minkowski inequality (see for example [START_REF] Schneider | Convex Bodies: The Brunn-Minkowski Theory[END_REF]Theorem 7.1.1]), one has:

r(tS r + (1 -t)U r ) = t × r(S r ) + (1 -t) × r(U r ) = t × r + (1 -t) × r = r and |tS r + (1 -t)U r | ≥ 1.
We then have:

∀t ∈ [0, 1], 1 r(K t r ) = |t × r(S r ) + (1 -t) × r(U r )| t × r(S r ) + (1 -t) × r(U r ) ≥ 1 r . ( 6
)
Figure 3: The continuous path Γ r .

Step 4: Stability of the paths:

For X = (x 1 , x 2 ) ∈ R 2 , we denote X = max(|x 1 |, |x 2 |).
In this step, we prove a continuity result on the paths Γ r r≤r(B) : let us take r 0 ∈ (0, r(B)] and ε > 0, we show that:

∃ α ε > 0, ∀r ∈ (r 0 -α ε , r 0 + α ε ) ∩ (0, r(B)], sup t∈[0,3] Γ r (t) -Γ r0 (t) ≤ ε. (7)
Let us take r ∈ [r 0 /2, r(B)], we have for every

t ∈ [1, 2]: Γ r (t) -Γ r0 (t) = max |x r,1 (t) -x r0,1 (t)|, |f 1 (x r,1 (t)) -f 1 (x r0,1 (t))| ≤ max |x r,1 (t) -x r0,1 (t)|, sup s∈[ √ π,+∞) |f 1 (s)| × |x r,1 (t) -x r0,1 (t)| = |x r,1 (t) -x r0,1 (t)| = (2 -t) 1 r - 1 r 0 ≤ 1 r × r 0 |r -r 0 | ≤ 2 r 2 0 |r -r 0 |.
With similar computations we obtain that for every t ∈ [2, 3], we have Γ r (t) -Γ r0 (t) ≤ 2 r0 |r -r 0 |. We then write:

sup t∈[1,3] Γ r (t) -Γ r0 (t) ≤ 2 r 2 0 |r -r 0 |. (8)
The remaining case (t ∈ [0, 1]) requires more computations. For every t ∈ [0, 1], we have

Γ r (t) -Γ r0 (t) ≤ max 2 r 2 0 , 128 r 6 0 Cr 0 >0 d H (K t r , K t r0 ).
Indeed, we used:

• for the term with inradii

1 r(K t r ) - 1 r(K t r0 ) = 1 r - 1 r 0 = 1 r × r 0 |r -r 0 | ≤ 2 r 2 0 |r -r 0 |,
• the first assertion of [14, Lemma 2.7] for the term with the Cheeger constants, with the sets K t p and K t p0 that we assume to contain the origin O and whose radial functions are denoted f p,t , f p0,t (if Ω is a convex set that contains the origin we denote f Ω : θ ∈ [0, 2π] -→ sup{λ ≥ 0, λ cos θ sin θ ∈ Ω} the radial function of Ω).

|h(K t r ) -h(K t r0 )| ≤ 2 min(r, r 0 ) 2 × f r,t -f r0,t ∞ (by the first assertion of [14, Lemma 2.7]) ≤ 8 r 2 0 × f p,t ∞ f p0,t ∞ min r(K t r ), r(K t r0 ) 2 × d H (K t r , K t r0 ) (by [6, Proposition 2]) ≤ 8 r 2 0 × d(K t r )d(K t r0 ) min(r, r 0 ) 2 × d H (K t r , K t r0 ) (because f r,t ∞ , f r0,t ∞ ≤) ≤ 32 r 4 0 × 4 r 2 0 × d H (K t r , K t r0 ) (because f r0,t ∞ ≤ f r0,t ∞ ≤ d(K t r0 ) and d(K t r0 ) ≤ 2 r 0 , see (9) below) = 128 r 6 0 × d H (K t r , K t r0 ),
where we used that for every Ω ∈ K 2 with unit area which contains a ball of radius r 0 /2, we have:

d(Ω) ≤ 1 2 P (Ω) ≤ |Ω| r(Ω) ≤ 2|Ω| r 0 = 2 r 0 . (9)
Moreover, we recall that h K t r 0 and h K t r respectively correspond to the support functions of the convex sets K t r0 and K t r , we have:

d H (K t r , K t r0 ) = h K t r -h K t r 0 ∞ = (1 -t)h Lr 0 + th Sr 0 |(1 -t)L r0 + tS r0 | - (1 -t)h Lr + th Sr |(1 -t)L r + tS r | ∞ ≤ (1 -t) h Lr 0 |(1 -t)L r0 + tS r0 | - h Lr |(1 -t)L r + tS r | ∞ +t h Sr 0 |(1 -t)L r0 + tS r0 | - h Sr |(1 -t)L p + tS p | ∞ ≤ 1 |(1 -t)L r + tS r | h Sr 0 -h Sr ∞ + h Lr 0 -h Lr ∞ + h Sr 0 ∞ + h Lr 0 ∞ 1 |(1 -t)L r + tS r | - 1 |(1 -t)L r0 + tS r0 | ≤ d H (S r0 , S r ) + d H (L r0 , L r ) + h Sr 0 ∞ + h Lr 0 ∞ × |(1 -t)L r + tS r | -|(1 -t)L r0 + tS r0 | ≤ d H (S r0 , S r ) + d H (L r0 , L r ) + h Sr 0 ∞ + h Lr 0 ∞ × 2 k=0 |W k (L r , S r ) -W k (L r0 , S r0 )| Gr 0 (r)
,

where W 1 , W 2 and W 3 are the classical Minkowski mixed volumes (we refer to [START_REF] Schneider | Convex Bodies: The Brunn-Minkowski Theory[END_REF] for definitions and more properties). Thus, we obtain:

sup t∈[0,1] Γ r (t) -Γ r0 (t) ≤ max 2 r 2 0 , 128 r 6 0 × G r0 (r). (10)
By combining ( 8) and ( 10), we write:

sup t∈[0,3] Γ r (t) -Γ r0 (t) ≤ max max 2 r 2 0 , 128 r 6 0 × G r0 (r), 2 r 2 0 |r -r 0 | -→ r→r0 0,
where the convergence G r0 (r) -→ r→r0 0 follows from the fact that d H (S r , S r0 ) -→ r→r0 0, d H (L r , L r0 ) -→ r→r0 0 and the continuity of the Minkowski mixed volumes W 1 , W 2 and W 3 for the Hausdorff distance (see [START_REF] Schneider | Convex Bodies: The Brunn-Minkowski Theory[END_REF]).

Finally, we deduce that lim

r→r0 sup t∈[0,3]
Γ r (t) -Γ r0 (t) = 0, which proves [START_REF] Brasco | Spectral inequalities in quantitative form[END_REF].

Step 5: Conclusion:

Now that we proved that the boundaries {(x, x + √ π) | x ≥ 1/r(B)} and {(x, x + π/x) | x ≥ 1/r(B)} are included in the diagram D, it remains to show that it is also the case for the set of points contained between them. We argue by contradiction, assuming that there exists A(x A , y A ) ∈ {(x, y) | x > 1/r(B) and x + π/x < y < x + √ π}, such that A / ∈ D. We consider the function φ A : x ∈ [1/r(B), +∞) -→ ind(Γ 1/x , A), where ind(Γ 1/x , A) is the index of A with respect to Γ 1/x (also called the winding number of the closed curve Γ 1/x around the point A).

• By Step 4 and the continuity of the index, the function φ A is constant on [1/r(B), +∞).

• By inequality (6), for every x > x A the point A is in the interior of Γ 1/x , thus φ A (x) = 0.

• On the other hand, the point A is in the exterior of

Γ 1/r(B) = {(1/r(B), 1/r(B)+ √ π)}, thus φ A (1/r(B)) = 0.
By the last three points we get a contradiction, thus A ∈ D. Finally, we get the equality

D = (x, y) | x ≥ x 0 and x + π x ≤ y ≤ x + √ π .
3 Improving the Cheeger inequality for planar convex sets

In this section, we provide the proof of Theorem 1.1 and prove some improved bounds for J in some special subclasses of K 2 , namely: triangles, rhombii and stadiums.

Proof of Theorem 1.1

Let Ω ∈ K 2 . We have by Hersch inequality [START_REF] Hersch | Sur la fréquence fondamentale d'une membrane vibrante: évaluations par défaut et principe de maximum[END_REF] and Faber-krahn inequaliy [START_REF] Faber | dass unter allen homogenen membranen von gleicher fl ache undgleicher spannung die kreisf örmige den tiefsten grundton gibt[END_REF][START_REF] Krahn | Über eine von rayleigh formulierte minimaleigenschaft des kreises[END_REF]:

|Ω|λ 1 (Ω) ≥ max πj 2 01 , π 2 |Ω| 4r(Ω) 2 .
On the other hand, we recall the upper estimate of Theorem 1.3:

|Ω|h(Ω) ≤ |Ω| r(Ω) + √ π.
Thus, we have:

J(Ω) = λ 1 (Ω) h(Ω) 2 ≥ max πj 2 01 , π 2 |Ω| 4r(Ω) 2 √ |Ω| r(Ω) + √ π 2 ≥ min x≥ √ π max πj 2 01 , π 2 x 2 4 (x + √ π) 2 = πj 01 2j 01 + π 2 ≈ 0.902... The minimum is taken over [ √ π, +∞) because √ |Ω| r(Ω) ≥ √ π for every Ω ∈ K 2 .
Moreover, it is attained for x = πj01 √ π , see Figure 4. We note that one can combine the following Protter's inequality [START_REF] Protter | A lower bound for the fundamental frequency of a convex region[END_REF]:

∀Ω ∈ K 2 , λ 1 (Ω) ≥ π 2 4 1 r(Ω) 2 + 1 d(Ω) 2 ,
which is an improvement of Hersch's inequality [START_REF] Hersch | Sur la fréquence fondamentale d'une membrane vibrante: évaluations par défaut et principe de maximum[END_REF]Section 8] with the optimal inequality (7) of [START_REF] Hernández Cifre | Some optimization problems for planar convex figures[END_REF]:

∀Ω ∈ K 2 , |Ω| r(Ω) 2 ≥ 1 + d(Ω) r(Ω) 2 + 2 arcsin 2r(Ω) d(Ω) := ϕ d(Ω) r(Ω) , (11)
to provide a slight improvement of the lower bound of Theorem 1.1. Indeed, the function ϕ is continuous and strictly increasing on [2, +∞) (we note that d(Ω)/r(Ω) ∈ [2, +∞), thus by considering the inverse function denoted ϕ -1 , inequality 11 becomes:

∀Ω ∈ K 2 , d(Ω) r(Ω) ≤ ϕ -1 |Ω| r(Ω) 2
. We then write:

λ 1 (Ω) h(Ω) 2 ≥ max   π 2 4r(Ω) 2 × 1 + 1 ϕ -1 |Ω| r(Ω) 2 2 , πj 2 01 |Ω|   1 r(Ω) + π |Ω| 2 = max   π 2 4 × |Ω| r(Ω) 2 ×   1 + 1 ϕ -1 |Ω| r(Ω) 2 2   , πj 2 01   |Ω| r(Ω) 2 1/2 + √ π 2 ≥ min x∈[π,+∞) max π 2 4 x 1 + 1 ϕ -1 (x) 2 , πj 2 01 ( √ x + √ π) (because |Ω| r(Ω) 2 ≥ πr(Ω) 2 r(Ω) 2 = π).
Numerical computations show that the latter minimum is approximately equal to 0.914..., which slightly improves the lower bound of Theorem 1.1.

Improvements for special classes of shapes

We provide some improved lower bounds of J for some special classes of planar convex sets. We note that the numerical bounds which appear in the following proposition are not exact.

Proposition 3.1. Let Ω ∈ K 2 .
1. If Ω is a triangle, then J(Ω) > 1.2076.

2.

If Ω is a rhombus, then J(Ω) ≥ 1.3819.

3.

If Ω is a stadium (i.e. the convex hull of two identical balls), then J(Ω) ≥ 1.3673.

Proof. Let Ω ∈ K 2 , since J is invariant by homothety (due to scaling properties of λ 1 and h), we may assume without loss of generality that |Ω| = 1.

1. Let us assume Ω to be a triangle and denote d its diameter and L its perimeter. To bound λ 1 (Ω) from below, we make use of two inequalities:

• The first one is the polygonal Faber-Krahn inequality for triangles, which states that between triangles of the same area, the regular one minimizes the first Dirichlet eigenvalue:

λ 1 (Ω) ≥ λ 1 (T eq ) = 4π 2 √ 3 ,
where T eq is the equilateral triangle of unit area (whose diameter is d eq = 2 3 1/4 ). • The second (more recent) is due to P. Freitas and B. Siudeja [13, Corollary 4.1]:

λ 1 (Ω) ≥ π 2 4|Ω| 2 d(Ω) + 2|Ω| d(Ω) 2 .
We then have on the one hand:

λ 1 (Ω) ≥ max 4π 2 √ 3 , π 2 4 d + 2 d 2 ,
and on the other hand, the Cheeger constant of the triangle Ω is given by:

h(Ω) = P (Ω) + 4π|Ω| 2|Ω| = L + √ 4π 2 ≤ L iso + √ 4π 2 ,
where L iso is the perimeter of the isoceles triangle whose diameter is equal to d and area equal to 1. By using Pythagoras' theorem, we have:

L iso = 2d + d -d 2 - 4 d 2 2 + 4 d 2 .
Finally, we obtain the following inequality:

J(Ω) ≥ φ 1 (d) := max 4π 2 √ 3 , π 2 4 d + 2 d 2   2d+ d-d 2 -4 d 2 2 + 4 d 2 + √ 4π 2   2 .
We note that d ≥ d eq . Indeed, by the isoperimetric inequality of triangles:

3d eq = L eq ≤ L ≤ 3d.
Numerically, we obtain min d≥deq φ 1 (d) ≈ 1.2076...

2.

Let us assume Ω to be the rhombus of unit area whose vertices are given by (-d/2, 0), (0, -1/d), (d/2, 0) and (0, 1/d).

We bound λ 1 (Ω) from below by using the following Hooker and Protter's estimate for for rhombi [START_REF] Hooker | Bounds for the first eigenvalue of a rhombic membrane[END_REF]:

λ 1 (Ω) ≥ π 2 d 2 + 1 d 2 .
As for the Cheeger constant, since Ω is a circumscribed polygon, we have:

h(Ω) = P (Ω) + 4π|Ω| 2|Ω| = 2 1 d 2 + d 2 4 + √ π
we use its explicit value in term of d.

J(Ω) ≥ φ 2 (d) = π 2 d 2 + 1 d 2 2 1 d 2 + d 2 4 + √ π 2 .
Numerically, we obtain min d≥ √ 2 φ 2 (d) ≈ 1.3819...

3.

Let us assume Ω to be a stadium of unit area whose diameter is given by a + 2r, where r > 0 is the radius of the ball of its extremity and a > 0. The condition |Ω| = 1 implies that πr 2 + 2ar = 1, which is equivalent to a = 1-πr 2 2r . We use the monotonicity of λ 1 for inclusion (for Ω ⊂ (-r, r) × (0, a + 2r)) and Faber-Krahn inequality to write:

λ 1 (Ω) ≥ max λ 1 (B), π 2 1 4r 2 + 1 (a + 2r) 2 = max λ 1 (B), π 2 4r 2 (1 + (4 -π)r 2 ) 2 + 1 4r 2 .
It is classical that the stadiums are Cheeger of themselves, see [START_REF] Kawohl | Characterization of Cheeger sets for convex subsets of the plane[END_REF], we then have:

h(Ω) = P (Ω) |Ω| = 2a + 2πr = 1 + πr 2 r .
Then:

J(Ω) ≥ φ 3 (r) := max λ 1 (B), π 2 4r 2 (1+(4-π)r 2 ) 2 + 1
β n := inf Ω∈K n J(Ω) ≤ inf ω∈K n-1 J(ω) =: β n-1 .
The idea is to prove that for any ω ∈ K n-1 , there exists a family (Ω d ) d>0 of elements of K n such that:

J(ω) = lim d→+∞ J(Ω d ).
The proof is decomposed in 3 steps.

Step 1: Lower estimates for λ 1 and h

Let us take Ω ∈ K n . We can assume without loss of generality that inf{t ∈ R | Ω ∩ {x 1 = t} = ∅} = 0 and denote

M Ω := sup{t ∈ R | Ω ∩ {x 1 = t} = ∅}.
We introduce the functions

φ λ : t ∈ [0, M Ω ] -→          λ 1 (Ω ∩ {x 1 = t}) ∈ R * + if t ∈ (0, M Ω ), λ 1 (Ω ∩ {x 1 = t}) ∈ R * + if t ∈ {0, M Ω } and |Ω ∩ {x 1 = t}| > 0, +∞ if t ∈ {0, M Ω } and |Ω ∩ {x 1 = t}| = 0, and
φ h : t ∈ [0, M Ω ] -→          h(Ω ∩ {x 1 = t}) ∈ R * + if t ∈ (0, M Ω ), h(Ω ∩ {x 1 = t}) ∈ R * + if t ∈ {0, M Ω } and |Ω ∩ {x 1 = t}| > 0, +∞ if t ∈ {0, M Ω } and |Ω ∩ {x 1 = t}| = 0, Since the function t ∈ (0, M Ω ) -→ Ω ∩ {x 1 = t} ∈ (K n-1 , d H
) is continuous and λ 1 and h are continuous for the Hausdorff distance d H on K n-1 , we have that the functions φ λ and φ h are continuous on (0, M Ω ). We then distinguish the cases "φ λ (0) = +∞ or φ λ (M Ω ) = +∞" (resp. "φ h (0) = +∞ or φ h (M Ω ) = +∞") and "φ λ (0) ∈ R and φ λ (M Ω ) ∈ R" (resp. "φ h (0) ∈ R and φ h (M Ω ) ∈ R") and use the intermediate values theorem to show that there exist t λ , t h ∈ [0, M Ω ] such that min t∈(0,MΩ) φ λ (t) = φ λ (t λ ) and min t∈(0,MΩ)

φ h (t) = φ h (t λ ).
In the proof of [START_REF] Brasco | Spectral inequalities in quantitative form[END_REF]Lemma 7.63], the authors prove that:

λ 1 (Ω) ≥ λ 1 (Ω ∩ {x 1 = t λ }).
Let us prove a similar result for Cheeger's constant (i.e. h(Ω) ≥ h(Ω ∩ {x 1 = t h }).). We have

h(Ω) = P (C Ω ) |C Ω | = ´d 0 P (C Ω ∩ {x 1 = t})dt ´d 0 |C Ω ∩ {x 1 = t}|dt ≥ ´d 0 h(Ω ∩ {x 1 = t})|C Ω ∩ {x 1 = t}|dt ´d 0 |C Ω ∩ {x 1 = t}|dt ≥ h(Ω ∩ {x 1 = t h }) ´d 0 |C Ω ∩ {x 1 = t}|dt ´d 0 |C Ω ∩ {x 1 = t}|dt = h(Ω ∩ {x 1 = t h }).
Step 2: Study of sets with increasing diameters and fixed volume

Let (Ω k ) a sequence of elements K n of the same volumes 1, such that d k := d(Ω k ) → +∞.
Let us prove that:

lim inf k→+∞ J(Ω k ) ≥ inf ω∈K n-1 J(ω).
For every k ∈ N, we consider A k and A k two diametrical points of Ω k (ie. such as

|A k A k | = d k ).
Since J is invariant by rigid motions we can assume without loss of generality that A k = (0, ..., 0) and A k = (d k , 0, ..., 0).

By

Step 1, we have for all k ∈ N:

λ 1 (Ω k ) ≥ λ 1 (ω k )
where

ω k := Ω k ∩ {x = t k }.
We can assume without loss of generality that t k ≥ d k /2. Let T k be the cone obtained by taking the convex hull of {A k } ∪ C k , where C k is the Cheeger set of the convex section ω k .

Let α ∈]0, 1[, we introduce the tube U α k := αC k × (0, (1 -α)t k ). By convexity, we have the following inclusions:

U α k ⊂ T k ⊂ Ω k
By definition of the Cheeger constant, we have:

h(Ω k ) ≤ P (U α k ) |U α k | = 2α n-1 |C k | + α n-2 (1 -α)P (C k )t k α n-1 (1 -α)|C k |t k = 2 (1 -α)t k + h(ω k ) α ∼ k→+∞ h(ω k ) α .
Indeed:

1 t k = o k→∞ P (C k ) |C k |
, because:

|C k | P (C k ) = |C k | P |C k | 1 n × C k |C k | 1 n = |C k | |C k | n-1 n P C k |C k | 1 n = |C k | 1 n P C k |C k | 1 n ≤ |C k | 1 n P (B n-1 ) ≤ |Ω k | 1 n n 1 n P (B n-1 ) × 1 t 1/n k = o k→∞ (t k ),
where

B n-1 ⊂ R n-1 is a ball of volume 1.
We deduce that:

∀α ∈ (0, 1), J(Ω k ) ≥ λ 1 (ω k ) 2 (1-α)t k + h(ω k ) α 2 ∼ k→+∞ α 2 J(ω k ) Thus: ∀α ∈ (0, 1), lim inf k→+∞ J(Ω k ) ≥ α 2 lim inf k→+∞ J(ω k ) ≥ α 2 inf ω∈K n-1 J(ω)
By letting α → 1, we obtain:

lim inf k→+∞ J(Ω k ) ≥ inf ω∈K n-1 J(ω)
Step 3: Study of long tubes

In this step, we show that when the height of a tube goes to infinity, the value of J of this tube converges to the value corresponding to the (n -1)-dimensional section given by its basis. More precisely, if we take ω ∈ K n-1 , we prove that

lim d→+∞ J [0, d] × ω = J(ω).
We have by Step 2:

lim inf k→+∞ J(Ω k ) ≥ J(ω) It is classical that λ 1 [0, d] × ω = π d 2 + λ 1 (ω),
which follows from the use the separation of variables and the orthogonality of Laplacian eigenfunctions.

As for the Cheeger constant, we use the result of Step 1:

h [0, d] × ω ≥ h(ω).
Thus, we have:

J [0, d] × ω = λ 1 [0, d] × ω h [0, d] × ω 2 ≤ π d 2 + λ 1 (ω) h(ω) 2 .
By passing to superior limit:

lim sup d→+∞ J [0, d] × ω ≤ J(ω).
Then:

lim d→+∞ J [0, d] × ω = J(ω).
At last, we write:

β n-1 = inf ω∈K n-1 J(ω) = inf ω∈K n-1 lim d→+∞ J [0, d] × ω ≥ inf Ω∈K n J(Ω) ≥ inf Ω∈K n J(Ω) = β n .
2. For every n ≥ 2, we take a ball B n ⊂ R n of unit radius, we have:

1 4 ≤ inf Ω∈K n J(Ω) ≤ J(B n ) = λ 1 (B n ) h(B n ) 2 = j 2 n 2 -1,1 n 2 ∼ n→+∞ n 2 2 n 2 = 1 4 ,
where j n 2 -1,1 is the first root of the n th Bessel function of first kind. We refer to [START_REF] Tricomi | Sulle funzioni di Bellel di ordine e argomento pressochè uguali[END_REF] for the equivalence

j n 2 -1,1 ∼ n→+∞ n 2 .

The existence result:

Now, we assume that: inf Ω∈K n J(Ω) < inf ω∈K n-1 J(ω). Let us prove the existence of a minimizer of J on K n . Let (Ω k ) be a minimizing sequence of K n (ie. such as

lim k→+∞ J(Ω k ) = inf Ω∈K n J(Ω)
). Since J is scaling invariant we can assume without loss of generality that |Ω k | = 1 for all n ∈ N.

If d(Ω k ) is not bounded, we can extract a subsequence (Ω ϕ(k) ) such as lim k→+∞ d(Ω ϕ(k) ) = +∞.
Thus, by Step 2:

inf Ω∈K n J(Ω) = lim k→+∞ J(Ω k ) ≥ inf ω∈K n-1 J(ω). which contradicts hypothesis β n-1 > β n .
We deduce that the sequence of diameters d(Ω k ) is bounded, then by compactness, there exists Ω * ∈ K n and a strictly increasing map σ : N -→ N such that Ω σ(k) -→ k→∞ Ω * for Hausdorff distance. We then have by continuity of J for the same metric (see [START_REF] Parini | Reverse Cheeger inequality for planar convex sets[END_REF]Proposition 3.2]):

J(Ω * ) = lim k→+∞ J(Ω k ) = inf Ω∈K n J(Ω).
4.2 Discussion of the hypothesis β n < β n-1

We believe that hypothesis β n < β n-1 is true for any dimension n and that one can use convex cylinders (i.e. those of the form ω × [0, d], where ω ∈ K n-1 and d > 0) to show it.

Let us analyse what happens when n = 2. In this case convex cylinders are rectangles. We consider the family of cylinders Ω d = [0, 1] × [0, d] (where d > 0) and denote

Ψ [0,1] : d > 0 -→ J(Ω d ) = λ 1 (Ω d ) h(Ω d ) 2 = π 2 1 + 1 d 2 4-π 1+d- √ (d-1) 2 +πd 2 .
We plot the curve of Ψ [0,1] in Figure 5. We note that for sufficiently high values of d the function Ψ [0,1] : d > 0 -→ J(Ω d ) is strictly increasing and converges to J (0, 1) = π

Ψ [0,1] : d > 0 -→ J [0, 1] × [0, d] .
We believe that the monotonicity property should hold in higher dimensions: let n ≥ 2, Ω d := ω × [0, d] where d > 0 and ω ∈ K n-1 , as before we denote Ψ ω : d > 0 -→ J(Ω d ). We have already proved above that:

lim d→+∞ Ψ ω (d) = lim d→+∞ J(Ω d ) = J(ω).
It remains to prove that for large values of d one has:

Ψ ω (d) = J(Ω d ) < J(ω).
To do so, we propose to show that function Ψ ω is strictly increasing for large values of d by studying the derivative Ψ ω (d).

Let us take d > 0, we have for t > 0 sufficiently small:

λ 1 (Ω d+t ) = λ 1 (ω) + π 2 (d + t) 2 = λ 1 (ω) + π 2 d 2 - 2π 2 d 3 t + o t→0 (t), and
h(Ω d+t ) = h(Ω d ) + h (Ω d , V d ) × t + o t→0 (t), where V d : R n → R n is the smooth dilatation field such that V d (x 1 , • • • , x n ) = 0, • • • , 0, xn d .
As proved in [START_REF] Parini | Shape derivative of the cheeger constant[END_REF], we have:

h (Ω d , V d ) = 1 |C Ω d | ˆ∂Ω d ∩∂CΩ d κ -h(Ω d ) V d , n dσ,
where κ is the mean curvature and C Ω d is the Cheeger set of Ω d . Since V d , n = 0 on all ∂Ω d ∩ ∂C Ω d except on the upper basis ∂Ω d ∩ ∂C Ω d ∩ {x n = d} where κ is null, we have the following formula for the shape derivative:

h (Ω d , V d ) = - |∂C Ω d ∩ {x n = d}| |C d | h(Ω d ).
By straightforward computations we obtain:

Ψ ω (d) = 1 h(Ω d ) 2 - 2π 2 d 3 + 2 λ 1 (ω) + π 2 d 2 |∂C Ω d ∩ {x n = d}| |C d | > 2π 2 h(Ω d ) 2 - 1 d 3 + λ 1 (ω) |∂C Ω d ∩ {x n = d}| |C d | ≥ 2π 2 d × h(Ω d ) 2 λ 1 (ω) |ω| |∂C Ω d ∩ {x n = d}| - 1 d 2 (because C Ω d ⊂ Ω d , thus |C Ω d | ≤ |Ω d | = |ω| × d).
Finally, it remains to prove that for sufficiently large values of d one can prove estimate of the type:

|∂C Ω d ∩ {x n = d}| > 1 d 2 .
One can check that this assertion is correct when n = 2. Indeed, if we consider the cylinder (rectangle)

Ω d = [0, 1] × [0, d],
we use the explicit expression of the Cheeger constant of rectangles that can be found in [START_REF] Kawohl | Characterization of Cheeger sets for convex subsets of the plane[END_REF]:

h(Ω d ) = 4 -π d + 1 -(d -1) 2 + πd = 1 + 1 d + 1 + π -2 d = 2 + π -1 2d + o d→+∞ 1 d .
Thus, as shown in Figure 6, we have:

|∂C Ω d ∩ {x 2 = d}| = 1 - 2 h(Ω d ) = π -1 4d + o d→+∞ 1 d ,
which proves that At last, let us mention the very recent work of E. Parini and V. Bobkov [START_REF] Bobkov | On the cheeger problem for rotationally invariant domains[END_REF] where they manage to explicitly describe the Cheeger sets of rationally invariant sets in any dimension and thus compute their Cheeger values. By applying these results to cylinders of the form B n-1 × [0, d], where B n-1 ⊂ R n-1 is a ball, we remark as expected that Ψ Bn-1 is strictly increasing for higher values of d and thus converges to J(B n-1 ) from below, which supports our strategy.

|∂C Ω d ∩ {x 2 = d}| ∼ d→+∞ π -1 4d > 1 d 2 .

Appendix: Some applications

In this Appendix, we apply the sharp estimates given in (4) to obtain some new bounds for the first Dirichlet eigenvalue in the case of planar convex sets.

5.1 Some sharp upper bounds for the first Dirichlet eigenvalue 5.1.1 General planar convex sets Proposition 5.1. We have the following sharp inequality:

∀Ω ∈ K 2 , λ 1 (Ω) < π 2 4 1 r(Ω) + π |Ω| 2 , ( 12
)
where equality is asymptotically attained by any family of convex sets (Ω k ) k∈N such as

|Ω k | = V 0 for any k ∈ N (where V 0 is a positive constant) and d(Ω k ) -→ k→+∞ +∞.
Proof. We have for every Ω ∈ K 2 :

λ 1 (Ω) < π 2 4 h(Ω) 2 ≤ π 2 4 1 r(Ω) + π |Ω| 2 ,
where the first inequality is the reverse Cheeger inequality (also called Buser inequality) proved by E. Parini in [27, Proposition 4.1] and the second inequality corresponds to the upper bound given in (4).

Let us now prove the sharpness inequality [START_REF] Faber | dass unter allen homogenen membranen von gleicher fl ache undgleicher spannung die kreisf örmige den tiefsten grundton gibt[END_REF]. Let V 0 > 0 and (Ω k ) k∈N a family of convex sets such as

|Ω k | = V 0 for any k ∈ N and d(Ω k ) -→ k→+∞ +∞.
We have on the one hand:

∀k ∈ N * , π 2 4r(Ω k ) 2 < λ 1 (Ω k ) < π 2 4 1 r(Ω k ) + π |Ω k | 2 = π 2 4 1 r(Ω k ) + π V 0 2 ,
on the other hand, we have:

1 r(Ω k ) ≥ P (Ω k ) 2|Ω k | ≥ d(Ω k ) V 0 -→ k→+∞ +∞, thus: λ 1 (Ω k ) ∼ k→+∞ π 2 4r(Ω k ) 2 , (13)
which proofs the sharpness of inequality [START_REF] Faber | dass unter allen homogenen membranen von gleicher fl ache undgleicher spannung die kreisf örmige den tiefsten grundton gibt[END_REF].

Remark 5.2. We note that one can use inequalities (4), to provide a similar equivalence as (13) for the Cheeger constant. Indeed, let us consider V 0 > 0 and (Ω k ) k∈N a family of convex sets such as |Ω k | = V 0 for any k ∈ N and d(Ω k ) -→ k→+∞ +∞. We have by (4):

∀k ∈ N, 1 r(Ω k ) + πr(Ω k ) V 0 ≤ h(Ω k ) ≤ 1 r(Ω k ) + π |V 0 | ,
and since 1 r(Ω k ) -→ k→+∞ +∞, we have the following equivalence:

h(Ω k ) ∼ k→+∞ 1 r(Ω k ) . ( 14
)
By combining (13) and (14), we retrieve (with an alternative method) the asymptotic result of [27, Proposition 4.1]:

lim k→+∞ J(Ω k ) = lim k→+∞ λ 1 (Ω k ) h(Ω k ) 2 = π 2 4 .
It is interesting to compare inequality [START_REF] Faber | dass unter allen homogenen membranen von gleicher fl ache undgleicher spannung die kreisf örmige den tiefsten grundton gibt[END_REF] with other inequalities involving the inradius and the area. One immediate estimate can be obtained by considering the inclusion B r(Ω) ⊂ Ω (where B r(Ω) is an inscribed ball of Ω (with radius r(Ω)). We have by the monotonicity of λ 1 :

λ 1 (Ω) ≤ λ 1 B r(Ω) = j 2 01 r(Ω) 2 , (15)
where j 01 denotes the first zero of the first Bessel function. This inequality was already stated in [27, inequality (3)] and in [8, inequality (1.5)] in higher dimensions and for a more general setting. In Figure 7, we plot the curves corresponding to the latter inequalities and an approximation of the Blaschke-Santaló diagram corresponding to the functionals λ 1 , the inradius r and the area | • |, obtained by generating 10 4 random convex sets. The diagram corresponds to the set of points: 12) improves ( 15) for convex sets with small inradius (ie. large 1 r ).

D := 1 r(Ω) , λ 1 (Ω) | Ω ∈ K 2 and |Ω| = 1 .
Remark 5.3. We note that the use of numerical simulations can be very helpful to have an idea on the shape and the properties of Blaschke-Santaló diagrams and state possible conjectures. It is then common to generate a large number of random domains, compute the values of the involved functionals and then obtain a cloud of dots that approximates the Blaschke-Santaló diagram. For various examples we refer to [START_REF] Antunes | New bounds for the principal Dirichlet eigenvalue of planar regions[END_REF][START_REF] Ftouhi | Complete systems of inequalities relating the perimeter, the area and the Cheeger constant of planar domains[END_REF][START_REF] Ftouhi | Blaschke-santaló diagram for volume, perimeter, and first dirichlet eigenvalue[END_REF][START_REF] Lucardesi | On blaschke-santaló diagrams for the torsional rigidity and the first dirichlet eigenvalue[END_REF]..

5.1.2

Sets that are homothetic to their form bodies: in particular "triangles"

We recall that in the case of sets that are homothetic to their form bodies, one has 1 2 P (Ω)r(Ω) = |Ω| and:

h(Ω) = P (Ω) + 4π|Ω| 2|Ω| = 1 r(Ω) + π |Ω| .
Thus one can write the following result, which is an immediate Corollary of the reverse Cheeger's inequality of [START_REF] Parini | Reverse Cheeger inequality for planar convex sets[END_REF]Proposition 4.1]:

Corollary 1. For every set Ω ∈ K 2 , that is homothetic to its form body (in particular triangles), we have the following inequality:

λ 1 (Ω) < π 2 4 × 1 r(Ω) + π |Ω| 2 = π 2 16 × P (Ω) |Ω| + 2 π |Ω| 2 . ( 16
)
The inequality is sharp as it is asymptotically attained by any sequence of convex sets (Ω k ) of unit area that are homothetic to their form bodies such that d(Ω k ) -→ k→+∞ +∞.

The most important thing about this upper bound is that in the case of triangles, inequality [START_REF] Henrot | Shape variation and optimization[END_REF] is better than the following bound obtained by B. Siudeja in [START_REF] Siudeja | Sharp bounds for eigenvalues of triangles[END_REF]Theorem 1.1] for "thin" triangles:

λ 1 (T ) ≤ π 2 9 × P (T ) |T | 2 . ( 17
)
It is also interesting to note that inequality ( 16) is even better (also for thin triangles) than the following upper bound stated in [32, Conjecture 1.2]:

Conjecture 1. For every triangle T , one has:

λ 1 (T ) ≤ π 2 12 × P (T ) |T | 2 + √ 3π 2 3|T | . ( 18
)
Here also, let us compare the different estimates in a Blaschke-Santaló diagram: we consider the one involving the perimeter, the area and λ 1 in the class of triangles, that is the set of points: 16) and [START_REF] Hernández Cifre | Is there a planar convex set with given width, diameter, and inradius?[END_REF] and Conjecture [START_REF] Hernández Cifre | Optimizing the perimeter and the area of convex sets with fixed diameter and circumradius[END_REF] with a zoom on smaller values of the perimeter.

A sharp Cheeger-type inequality

Proposition 5.4. We have the following sharp Cheeger-type inequality:

∀Ω ∈ K 2 , λ 1 (Ω) > π 2 4 h(Ω) - π |Ω| 2 , ( 19
)
where equality is asymptotically attained by any family of convex sets (Ω k ) k∈N such as |Ω k | = V 0 for any k ∈ N (where V 0 is a positive constant) and d(Ω k ) -→ k→+∞ +∞.

Proof. Let Ω ∈ K 2 , we have:

λ 1 (Ω) > π 2 4 × 1 r(Ω) 2 ≥ π 2 4 h(Ω) - π |Ω| 2 ,
where the first inequality is the classical Hersch's inequality [START_REF] Hersch | Sur la fréquence fondamentale d'une membrane vibrante: évaluations par défaut et principe de maximum[END_REF] and the second follows from is the upper estimate of (4). We note that inequality [START_REF] Hernández Cifre | Some optimization problems for planar convex figures[END_REF] is better than the improved Cheeger inequality of Theorem 1.1 (and even the conjecture J(Ω) ≥ J((0, 1) 2 )) for thin planar convex domains, see Figure 9, where we provide an approximation of the following Blaschke-Santaló diagram relating λ 1 , the Cheeger constant and the area:

  x∈Ωx, u the support function of the convex body Ω, it is the function that describes the distance from the origin to the supporting hyperplane of Ω with normal u.

4 4. 5 5Figure 1 :

 451 Figure 1: The diagram of the triplet (r, h, | • |).

Figure 2 :

 2 Figure 2: Idea of proof of the upper bound, with Ω = [-1, 1] × [0, 10].

  thus by continuity of the inradius and the Cheeger constant for the Hausdorff distance (see the proof of [11, Proposition 1] and [27, Proposition 3.1

Figure 4 : 2 01 , π 2 x 2 4 (x+ √ π) 2 . 3 . 2 A

 424232 Figure 4: Curve of the function x -→ max πj 2 01 , π 2 x 2 4 (x+ √ π) 2

4r 2 1+πr 2 r 2 Numerically, we obtain min r∈ 0, 1 √ π φ 3 2 1.

 2132 (r) ≈ 1.3673... 4 On the existence of a minimizer in higher dimensions 4.1 Proof of Theorem 1.Let n ≥ 2, let us first prove that:

Figure 5 :

 5 Figure 5: Curve of the functionΨ [0,1] : d > 0 -→ J [0, 1] × [0, d] .

Figure 6 :

 6 Figure 6: Cheeger set of the rectangle Ω d .

Figure 7 :

 7 Figure 7: Inequality (12) improves (15) for convex sets with small inradius (ie. large 1 r ).

TFigure 8 :

 8 Figure 8: Comparison between inequalities (16) and[START_REF] Hernández Cifre | Is there a planar convex set with given width, diameter, and inradius?[END_REF] and Conjecture[START_REF] Hernández Cifre | Optimizing the perimeter and the area of convex sets with fixed diameter and circumradius[END_REF] with a zoom on smaller values of the perimeter.

2 4

 2 As for the equality case, let (Ω k ) a family of convex sets (Ω k ) k∈N such as |Ω k | = V 0 for any k ∈ N and d(Ω k ) -→ k→+∞ +∞. By [27, Proposition 4.1], we have: λ 1 (Ω k ) ∼ k→+∞ π h(Ω k ) 2 and by the equivalence (14) and lim k-→+∞ 1 r(Ω k ) = +∞ (see the proof of Proposition 5.1), we have lim k-→+∞ h(Ω k ) = +∞ which implies the equivalence: λ 1 (Ω) ∼

C

  := h(Ω), λ 1 (Ω) | Ω ∈ K 2 and |Ω| = 1 .

Figure 9 :

 9 Figure 9: Approximation of the Blaschke-Santaló diagram C and relevant inequalities.

Acknowledgments

The author would like to thank Jimmy Lamboley for pointing out this problem to his attention. The author would also like to thank Antoine Henrot, Jimmy Lamboley and Enea Parini for stimulating conversations and valuable comments that helped to improve the following manuscript. The author would also like to thank the anonymous referee for his careful reading and valuable comments.

This work was partially supported by the project ANR-18-CE40-0013 SHAPO financed by the French Agence Nationale de la Recherche (ANR).