Yves Martinez-Maure

Uniqueness results for the Minkowski problem extended to hedgehogs by

Keywords: Minkowski problem, Monge-Ampère equations, convex surfaces, hedgehogs MSC2010. 35A02, 35M10, 53A05, 53C45

The classical Minkowski problem has a natural extension to hedgehogs, that is to Minkowski differences of closed convex hypersurfaces. This extended Minkowski problem is much more difficult since it essentially boils down to the question of solutions of certain Monge-Ampère equations of mixed type on the unit sphere S n of R n+1 . In this paper, we mainly consider the uniqueness question and give first results.

Introduction

The classical Minkowski problem is that of the existence, uniqueness and regularity of closed convex hypersurfaces of the Euclidean linear space R n+1 whose Gaussian curvature (in the sense of Gauss' definition) is prescribed as a function of the outer normal vector. In the last century, this fundamental problem played an important role in the development of the theory of elliptic Monge-Ampère equations. Indeed, for C 2 + -hypersurfaces (C 2 -hypersurfaces with positive Gaussian curvature), this Minkowski problem is equivalent to the question of solutions of certain Monge-Ampère equations of elliptic type on the unit sphere S n of R n+1 .

Using approximation by convex polyhedra, Minkowski proved the existence of a weak solution [START_REF] Minkowski | Volumen und Oberfläche[END_REF]: If K is a continuous positive function on S n satisfying the following integral condition

S n u K(u) dσ(u) = 0,
where σ is the spherical Lebesgue measure on S n , then K is the Gaussian curvature of a unique (up to translation) closed convex hypersurface H. The uniqueness comes from the equality condition in a Minkowski's inequality (e.g. [18, p. 397]). The strong solution is due to Pogorelov [START_REF] Pogorelov | The Minkowski multidimensional problem[END_REF] and Cheng and Yau [START_REF] Cheng | On the regularity of the solution of the ndimensional Minkowski problem[END_REF] who proved independently that: if K is of class C m , (m ≥ 3), then the support function of H is of class C m+1,α for every α ∈ ]0, 1[. This classical Minkowski problem has a natural extension to hedgehogs, that is to Minkowski differences H = K -L of closed convex hypersurfaces K, L ∈ R n+1 , at least if we restrict ourselves to hypersurfaces whose support functions are C 2 . Indeed, the inverse of the Gaussian curvature of such a hedgehog is well-defined and continuous all over S n (see Section 1), so that the following existence question arises naturally:

(Q 1 )
Existence of a C 2 -solution: What are necessary and sufficient conditions for a real continuous function R ∈ C (S n ; R) to be the curvature function (that is, the inverse 1 K of the Gaussian curvature K) of some hedgehog H = K -L ?

Now let us expound the uniqueness question. As we shall see later, for any h ∈ C 2 (S 2 ; R), -h and h are the respective support functions of two hedgehogs H -h and H h of R 3 that have the same curvature function and are such that

H -h = s (H h ) ,
where s is the symmetry with respect to the origin of R 3 . Here, we have to recall that noncongruent hedgehogs of R 3 may have the same curvature function [START_REF] Martinez-Maure | New notion of index for hedgehogs of R 3 and applications[END_REF]: for instance, the two smooth (but not analytic) functions f , g defined on S 2 by

f (u) :=    exp(-1/z 2 ) if z = 0 0 if z = 0 and g (u) :=    sign (z) f (u) if z = 0 0 if z = 0,
where u = (x, y, z) ∈ S 2 , are the support functions of two noncongruent hedgehogs H f and H g of R 3 having the same curvature function R := 1/K ∈ C (S 2 ; R), (cf. Figure 1).

Figure 1. Noncongruent hedgehogs with the same curvature function

Consequently, we state the uniqueness question as follows:

(Q 2 ) Uniqueness of a C 2 -solution: If R ∈ C (S n ; R)
is the curvature function of some hedgehog H, what necessary and sufficient additional conditions must it satisfy in order that H be the unique hedgehog of which R is the curvature function (up to an isometry of the space) ?

In particular, it would be very interesting to know whether there exists any pair of noncongruent analytic hedgehogs of R 3 with the same curvature function (by 'analytic hedgehogs' we mean 'hedgehogs with an analytic support function'). We shall see in Section 1 that this latest question presents similarities to the open question of knowing whether there exists any pair of noncongruent isometric analytic closed surfaces in R 3 .

For n = 1, the problem is linear and so can be solved without difficulty [START_REF] Martinez-Maure | Geometric study of Minkowski differences of plane convex bodies[END_REF]. But for n = 2, the problem is already very difficult: if R ∈ C (S 2 ; R) changes sign on S 2 , the question of existence, uniqueness and regularity of a hedgehog of which R is the curvature function boils down to the study of a Monge-Ampère equation of mixed type, a class of equations for which there is no global result but only local ones by Lin [START_REF] Lin | The local isometric embedding in R 3 of 2-dimensional Riemannian manifolds with nonnegative curvature[END_REF] and Zuily [START_REF] Zuily | Existence locale de solutions C ∞ pour des équations de Monge-Ampère changeant de type[END_REF]. In the present paper, we are mainly interested in the uniqueness question. Question (Q 2 ) is too difficult to be solved at the present time and our main purpose will be simply to provide conditions under which two hedgehogs of R 3 have distinct curvature functions.

Let H 3 be the R-linear space of C 2 -hedgehogs defined up to a translation in R 3 (by 'C 2 -hedgehogs' we mean 'hedgehogs with a C 2 support function'). Our first result will be the following.

Theorem. Let H and H ′ be C 2 -hedgehogs that are linearly independent in H 3 . If some linear combination of H and H ′ is of class C 2 + , then H and H ′ have distinct curvature functions.

As we shall recall in Section 2, any hedgehog can be uniquely split into the sum of its centered and projective parts. Our second result relies on this decomposition of hedgehogs into their centered and projective parts.

Theorem. Let H and H ′ be two C 2 -hedgehogs that are linearly independent in H 3 and the centered parts of which are nontrivial (i.e., distinct from a point) and proportional to one and the same convex surface of class C 2 + . Then H and H ′ have distinct curvature functions.

An immediate consequence will be that:

Corollary. Two C 2 -hedgehogs of nonzero constant width that are linearly independent in H 3 have distinct curvature function.

Our last result relies on the extension to hedgehogs of the notion of mixed curvature function, which will be recalled in Section 1.

Theorem. Let H and H ′ be analytic (resp. projective C 2 ) hedgehogs of R 3 that are linearly independent in H 3 . If the mixed curvature function of H and H ′ does not change sign on S 2 , then H and H ′ have distinct curvature functions.

In Section 1, we shall begin by recalling some basic definitions and facts and by presenting what is already known on the Minkowski problem extended to hedgehogs. Later, we shall see different ways of constructing pairs of noncongruent hedgehogs having the same curvature function.

Section 2 will be devoted to the statement of the main results and Section 3 to the proofs and further remarks.

Basic facts and observations on the extended Minkowski problem

As is well-known, every convex body K ⊂ R n+1 is determined by its support function h

K : S n -→ R, where h K (u) is defined by h K (u) = sup { x, u |x ∈ K }, (u ∈ S n ),
that is, as the signed distance from the origin to the support hyperplane with normal vector u. In particular, every closed convex hypersurface of class C 2 + (i.e., C 2 -hypersurface with positive Gaussian curvature) is determined by its support function h (which must be of class C 2 on S n [18, p. 111]) as the envelope H h of the family of hyperplanes with equation x, u = h(u). This envelope H h is described analytically by the following system of equations

x, u = h(u) x, . = dh u (.)
.

The second equation is obtained from the first by performing a partial differentiation with respect to u. From the first equation, the orthogonal projection of x onto the line spanned by u is h (u) u and from the second one, the orthogonal projection of x onto u ⊥ is the gradient of h at u (cf. Figure 2). Therefore, for each u ∈ S n , x h (u) = h(u)u + (∇h) (u) is the unique solution of this system.

Figure 2. Hedgehogs as envelopes parametrized by their Gauss map

Now, for any C 2 -function h on S n , the envelope H h is in fact well-defined (even if h is not the support function of a convex hypersurface). Its natural parametrization x h : S n → H h , u → h(u)u + (∇h) (u) can be interpreted as the inverse of its Gauss map, in the sense that: at each regular point x h (u) of H h , u is a normal vector to H h . We say that H h is the hedgehog with support function h (cf. Figure 3). Note that x h depends linearly on h. In fact, we can introduce a more general notion of hedgehogs by regarding hedgehogs of R n+1 as Minkowski differences of arbitrary convex bodies of R n+1 [START_REF] Martinez-Maure | Théorie des hérissons et polytopes[END_REF][START_REF] Martinez-Maure | Geometric study of Minkowski differences of plane convex bodies[END_REF]. But in the present paper, we shall only consider hedgehogs with a C 2support function and we will refer to them as 'C 2 -hedgehogs'.

Gaussian curvature of C 2 -hedgehogs

Let H n+1 denote the R-linear space of C 2 -hedgehogs defined up to a translation in the Euclidean linear space R n+1 and identified with their support functions. Analytically speaking, saying that a hedgehog H h ⊂ R n+1 is defined up to a translation simply means that the first spherical harmonics of its support function is not specified.

As we saw before, elements of H n+1 may be singular hypersurfaces. Since the parametrization x h can be regarded as the inverse of the Gauss map, the Gaussian curvature K h of H h at x h (u) is given by K h (u) = 1/ det [T u x h ], where T u x h is the tangent map of x h at u. Therefore, singularities are the very points at which the Gaussian curvature is infinite. For every u ∈ S n , the tangent map of x h at the point u is T u x h = h(u) Id T u S n + H h (u), where H h (u) is the symmetric endomorphism associated with the hessian of h at u. Consequently, if λ is an eigenvalue of the hessian of h at u then λ + h (u) is (up to the sign) one of the principal radii of curvature of H h at x h (u) and the so-called 'curvature function' R h := 1/K h can be given by

R h (u) = det [H ij (u) + h (u) δ ij ] , (1)
where δ ij are the Kronecker symbols and (H ij (u)) the Hessian of h at u with respect to an orthonormal frame on S n . The case n = 2. From (1), the curvature function

R h := 1/K h of H h ⊂ R 3 is given by R h = (λ 1 + h) (λ 2 + h) = h 2 + h∆ 2 h + ∆ 22 h,
where ∆ 2 denotes the spherical Laplacian and ∆ 22 the Monge-Ampère operator (respectively the sum and the product of the eigenvalues λ 1 , λ 2 of the Hessian of h). So, the equation we shall be dealing with will be the following

h 2 + h∆ 2 h + ∆ 22 h = 1/K.
Note that the so-called 'mixed curvature function' of hedgehogs of R 3 , that is,

R : H 2 3 → C (S 2 ; R) (f, g) → R (f,g) := 1 2 (R f +g -R f -R g )
is bilinear and symmetric:

(i) ∀ (f, g, h) ∈ H 3 3 , ∀λ ∈ R, R (f +λg,h) = R (f,h) + λR (g,h) ; (ii) ∀ (f, g) ∈ H 2 3 , R (g,f ) = R (f,g) . For any h ∈ H 3 , we have in particular R -h = R h . Note that R (1,f ) = 1 2 (∆ 2 h + 2h
) is (up to the sign) half the sum of the principal radii of curvature of H h ⊂ R 3 .

Nonexistence in the Minkowski problem for hedgehogs

The point is that the curvature function R h := 1/K h of any C 2 -hedgehog H h of R n+1 is well-defined and continuous all over S n , including at the singular points of x h , so that the Minkowski problem arises naturally for hedgehogs. In this paper, we are thus interested in studying the existence and/or uniqueness of C 2 -solutions to the Monge-Ampère equation

R h = R, (2)
where R is a given real continuous function on S n . As in the classical Minkowski problem, the following integral condition is necessary for the existence of such a solution:

S n R (u) udσ(u) = 0. (3)
It simply expresses that any

C 2 -hedgehog H h of R n+1 is a closed hypersurface.
But it is no longer sufficient: for instance, the constant function equal to -1 on S 2 satisfies integral condition (3) but it cannot be the curvature function of a hedgehog since there is no compact surface with negative Gaussian curvature in R 3 . This extended Minkowski problem leads to the following examples of Monge Ampère equations of mixed type for which there is no solution. For every v ∈ S 2 , the smooth function F v (u) = 1 -2 u, v 2 satisfies integral condition (3) but is not a curvature function on S 2 [START_REF] Martinez-Maure | Hedgehogs and zonoids[END_REF]. In other words, for every fixed v ∈ S 2 , the Monge Ampère equation h 2 + h∆ 2 h + ∆ 22 h = F v has no C 2 -solution on S 2 . The proof makes use of orthogonal projection techniques adapted to hedgehogs.

Nonuniqueness in the Minkowski problem for hedgehogs

As recalled in the introduction, two noncongruent hedgehogs of R 3 may have the same curvature function. By bilinearity and symmetry in the arguments of the mixed curvature function R : H 2 3 → C (S 2 ; R), if H f and H g are two hedgehogs of R 3 having the same curvature function then, for all (λ, µ) ∈ R 2 , the hedgehogs H λf +µg and H µf +λg also have the same curvature function. For instance, from the pair {H f , H g } of noncongruent hedgehogs represented in Figure 1, we deduce the pair {H f +2g , H 2f +g } of noncongruent hedgehogs (which have the same curvature function) represented in Figure 5. We can assume that S admits a parametrization of the form

x : S 2 → S ⊂ R 3 u → ρ (u) u,
where ρ is a smooth positive function. Then the hedgehog with support function h = 1/ρ can be regarded as the dual surface of S [START_REF] Martinez-Maure | Hedgehogs of constant width and equichordal points[END_REF]. This hedgehog H h is a surface of revolution whose generating curve (a plane hedgehog which has a fish form) is represented in Figure 7. Replacing the fish's tail by its image under the symmetry with respect to the double point (which by duality corresponds to the plane P ) and rotating the plane hedgehog that we get around its axis of symmetry, we generate an other hedgehog which has the same curvature function as H h without being congruent to it.

Statement of results

Recall that H 3 denotes the R-linear space of C 2 -hedgehogs defined up to a translation in R 3 . Our first result below will be a consequence of the classical Minkowski's uniqueness theorem.

Theorem 1. Let H f and H g be C 2 -hedgehogs that are linearly independent in H 3 . If some linear combination of H f and H g is of class C 2 + , then H f and H g have distinct curvature functions.

Our second result makes use of the decomposition of hedgehogs into their centered and projective parts.

Decomposition of a hedgehog into its centered and projective parts

Recall that a hedgehog H h of R n+1 is said to be centered (resp. projective) if its support function h is symmetric (resp. antisymmetric), that is, if we have:

∀u∈ S n , h (-u) = h (u) (resp. h (-u) = -h (u)) .
For instance, the hedgehog H f (resp. H g ) of R 3 that is represented in Figure 1.a (resp. Figure 1.b) is centered (resp. projective). Geometrically speaking, this means that H h is centrally symmetric with respect to the origin (resp. that any pair of antipodal points on the unit sphere S n correspond to a same point on the hypersurface

H h = x h (S n )).
Now, the support function h of H h ⊂ R n+1 can be uniquely split into the sum of its symmetric and antisymmetric parts f and g:

∀u∈ S n , h (u) = f (u) + g (u) where    f (u) = 1 2 (h (u) + h (-u)) g (u) = 1 2 (h (u) -h (-u))
.

Consequently A hedgehog H h of R n+1 is said to be of constant width if its centered part has a constant support function. In other words, a hedgehog H h of R n+1 is of constant width if the signed distance between the two cooriented support hyperplanes that are orthogonal to u ∈ S n does not depend on u, that is, if:

∃r ∈ R, ∀u ∈ S n , h(u) + h(-u) = 2r.
A straightforward consequence of Theorem 3 is the following corollary.

Corollary 3. Let H f and H g be C 2 -hedgehogs that are linearly independent in H 3 . If H f and H g are of nonzero constant width, then their curvature functions R f and R g are distinct.

A hedgehog H h of R n+1 is said to be analytic if its support function h is C ω on S n . Theorem 4. Let H f and H g be analytic (resp. projective C 2 ) hedgehogs of R 3 that are linearly independent in H 3 . If the mixed curvature function of H f and H g does not change sign on S 2 , then H f and H g have distinct curvature functions.

Proof of the results and further remarks

Proof of Theorem 1. By assumption, there exists (λ, µ) ∈ R 2 such that the hedgehog H λf +µg is of class C 2 + . We can assume that |λ| = |µ|. Let assume that R f = R g . We then have:

R λf +µg = λ 2 R f + µ 2 R g + 2λµR (f,g) = µ 2 R f + λ 2 R g + 2µλR (f,g) = R µf +λg .
As the hedgehog H λf +µg is of class C 2 + , the equality R λf +µg = R µf +λg implies the existence of an ε ∈ {-1, 1} such that λf + µg = ε (µf + λg) by Minkowski's uniqueness theorem. We thus have (λεµ) f = ε (λεµ) g and hence f = εg since λεµ = 0.

Lemma 5. Let H f and H g be two C 2 -hedgehogs of R 3 . If u ∈ S 2 is such that R g (u) > 0, then R (f,g) (u) 2 ≥ R f (u) R g (u) .
Splitting into symmetric and antisymmetric parts, we get

   R f 1 + λ 2 1 R k = R f 2 + λ 2 2 R k λ 1 R (f 1 ,k) = λ 2 R (f 2 ,k)
.

By linearity of the mixed curvature function in the first argument, the second equation is equivalent to

R (λ 1 f 1 -λ 2 f 2 ,k) = 0. By Lemma 7, this implies that H λ 1 f 1 -λ 2 f 2 is a point and hence that H λ 1 f 1 = H λ 2 f 2 in H 3
. Now, by multiplying each member of the first equation of the previous system by λ 2 1 , we get

λ 2 1 R f 1 + λ 4 1 R k = λ 2 1 R f 2 + λ 2 1 λ 2 2 R k ,
and hence

R λ 1 f 1 -R λ 1 f 2 = λ 2 1 λ 2 1 -λ 2 2 R k
by bilinearity of the mixed curvature function. Therefore,

R λ 2 f 2 -R λ 1 f 2 = λ 2 1 λ 2 1 -λ 2 2 R k , that is, λ 2 2 -λ 2 1 R f 2 -λ 2 1 R k = 0.
As H f 2 is projective (resp. H k is convex of class C 2 + ), we have [START_REF] Martinez-Maure | Indice d'un hérisson : étude et applications[END_REF]:

S 2 R f 2 dσ ≤ 0 and S 2 R k dσ > 0.
Therefore, R f 2 = λ 2 1 R k . From the previous equation, we thus get λ 2 2 = λ 2 1 , that is:

∃ε ∈ {-1, 1} , λ 2 = ελ 1 . Now, λ 1 H f 1 = H λ 1 f 1 = H λ 2 f 2 = λ 2 H f 2 and
H h 2 = H f 2 +λ 2 k = H f 2 + λ 2 H k = ε (H f 1 + λ 1 H k ) = εH h 1 in H 3 ,
which contradicts the fact that H h 1 and H h 2 are linearly independent in H 3 .

Lemma 8. Let H f and H g be two C 2 -hedgehogs of R 3 . If their curvature functions R f and R g are identically equal on S 2 , then either R f -g (u) ≤ 0 or R f +g (u) ≤ 0 for all u ∈ S 2 .

Proof of the lemma. Assume that R f -g (u) > 0 (resp. R f +g (u) > 0). By Lemma 5, we then have

R (f -g,f +g) (u) 2 ≥ R f -g (u) R f +g (u) . Now the assumption R f = R g implies R (f -g,f +g) = R f -R g = 0 and hence R f -g (u) R f +g (u) ≤ 0.
Therefore R f -g (u) ≤ 0 (resp. R f +g (u) ≤ 0).

Proof of Theorem 4. We prove the result by contraposition. Assume that R f and R g are identically equal on S 2 . Since H f and H g are analytic (resp. projective and C 2 ) hedgehogs of R 3 that are linearly independent in H 3 , it follows from Lemma 6 that there must exist (u, v) ∈ S 2 × S 2 such that R f -g (u) > 0 and R f +g (v) > 0. By Lemma 8, we then deduce that R f +g (u) ≤ 0 and R f -g (v) ≤ 0.

Now we have R (f,g) = 1 4 (R f +g -R f -g ), so that    R f -g (u) > 0 R f +g (u) ≤ 0 and    R f +g (v) > 0 R f -g (v) ≤ 0
implies R (f,g) (u) < 0 and R (f,g) (v) > 0.

Figure 3 .

 3 Figure 3. A hedgehog with a C 2 -support function Hedgehogs with a C 2 -support function can be regarded as the Minkowski differences of convex hypersurfaces (or convex bodies) of class C 2 + . Indeed [6], given any h ∈ C 2 (S n ; R), for all large enough real constant r, the functions h + r and r are support functions of convex hypersurfaces of class C 2 + such that h = (h + r)r.

Figure 4 .

 4 Figure 4. Hedgehogs as differences of convex bodies of class C 2 +

Figure 5 .

 5 Figure 5. Noncongruent hedgehogs with the same curvature function

Figure 6 .Figure 7 .

 67 Figure 6. Noncongruent isometric surfaces of revolution [3, p. 131]

  , any hedgehog H h of R n+1 has a unique representation of the form H f +H g , where H f and H g are respectively a centered and a projective hedgehog. We say that H f and H g are respectively the centered and the projective part of H h . Theorem 2. Let H h 1 and H h 2 be C 2 -hedgehogs that are linearly independent in H 3 and the centered parts of which are nontrivial (i.e., distinct from a point) and proportional to one and the same convex surface of class C 2 + . Then H h 1 and H h 2 have distinct curvature functions.

  λ 1 , λ 2 are nonzero. Therefore, we haveH f 1 = εH f 2 in H 3 , that is, H f 2 = εH f 1 and hence

Proof of the lemma. Define Q : R → R by Q (t) = R f +tg (u). By bilinearity and symmetry of the mixed curvature function, we have:

of the quadratic trinomial Q(t). On the one hand, from the assumption R g (u) > 0, it follows that Q(t) > 0 for any large enough t. On the other hand, there exists some

Surprisingly, there exist nontrivial (i.e., distinct from a point) hedgehogs of R 3 whose curvature function is nonpositive all over S 2 [START_REF] Martinez-Maure | Contre-exemple à une caractérisation conjecturée de la sphère[END_REF][START_REF] Panina | New counterexamples to A. D. Alexandrov's hypothesis[END_REF], which disproves a conjectured characterization of the 2-sphere [START_REF] Alexandrov | On uniqueness theorem for closed surfaces (Russian)[END_REF][START_REF] Koutroufiotis | On a conjectured characterization of the sphere[END_REF]. However, the support function of such a hedgehog cannot be neither analytic nor antisymmetric on S 2 :

only if H f is reduced to a single point, that is, only if f is the restriction to S 2 of a linear form on R 3 .

Proof of the lemma. Since H g is of class C 2 + , we have

by Lemma 5. From R (f,g) (u) = 0, we then deduce that R f ≤ 0 which implies the result by Lemma 6.

Proof of Theorem 2. By assumption, h 1 and h 2 are of the form   

, where λ 1 , λ 2 are nonzero real numbers, f 1 , f 2 the support functions of projective hedgehogs and k the support function of a centered convex surface of class C 2 + . Assume that R h 1 = R h 2 . By bilinearity and symmetry of the mixed curvature function, this gives


Cédric Lopez
email: lopez@lirmm.fr

Violaine Prince
email: prince@lirmm.fr

Mathieu Roche
email: mroche@lirmm.fr

Automatic titling of Articles Using Position and Statistical Information

This paper describes a system facilitating information retrieval in a set of textual documents by tackling the automatic titling and subtitling issue. Automatic titling here consists in extracting relevant noun phrases from texts as candidate titles. An original approach combining statistical criteria and noun phrases positions in the text helps collecting relevant titles and subtitles. So, the user may benefit from an outline of all the subjects evoked in a mass of documents, and easily find the information he/she is looking for. An evaluation on real data shows that the solutions given by this automatic titling approach are relevant.

Introduction

Web pages contain a multitude of information concerning many domains. Very often, the user has to supply heavy cognitive efforts to find the information he/she is looking for. For handicapped persons, while the access to Internet is a tremendous vector of integration in society, the localization of information remains complex. One of the key domains of web pages accessibility, such as defined by a standard proposed by handicap associations (W3C standard), concerns the titling (and subtitling) of web pages. The main goal is to increase the legibility of pages obtained from a search engine, where the relevance of results is often weak, disheartening readers, or to improve pages indexing, in order to obtain a better search. Besides, automatic titling can be integrated into diverse applications. For instance, it might help the editorial staff, proposing to the author of a given text, a segmented version, according to the issue tackled by [START_REF] Akrifed | Segmentation automatique des textes, l'exemple du logiciel tropes: Bilan et per-spectives= automatic texts segmentation, example of tropes software: assessment and prospects[END_REF][START_REF] Prince | Text segmentation based on document understanding for information retrieval[END_REF] and automatically titled. So, a new industrial application, based on automatic titling, would include the automatic generation of contents, saving time.

One of the major benefits of the system described in this paper, is to help the user in assimilating the semantic contents of a set of textual document. Another is to allow him/her to quickly find the relevant information. Applied to textual resources, the proposed approach consists in providing texts subjects by using the automatically generated titles, and so to facilitate information communication and localization. Titles determination requires to know titles morphosyntactic structure, as well as their associated subtitles. From some statistical studies, performed on data described in section 3, concerning morphosyntactic characteristics, we propose a two-stages process. The main idea is to extract, from a given text, the most relevant noun phrase and use it as title. The first stage consists in extracting all noun phrases existing in the text (section 4.1). The second stage determines the most relevant phrase among those previously extracted (section 4.2). An evaluation, performed by human judgment on real data, is presented (section 5) and discussed. Experiments have been run on French data, but could be easily transposed to several Western languages, which share with French a rather common set of linguistic features (i.e., most Indo-European languages).

Previous Works

It seems that no scientific study leading to an automatic titling application was published. However, the title issue is studied in numerous works. Titling is a process aiming at relevantly representing the contents of documents. It might use metaphors, humor or emphasis, thus separating a titling task from a summarization process, proving the importance of rhetorical status in both tasks [START_REF] Teufel | Sentence extraction and rhetorical classification for flexible abstracts[END_REF]. Titles have been stud-ied as textual objects focusing on fonts, sizes, colors, [START_REF] Ho-Dac | Sur la fonction discursive des titres[END_REF]. Also, since a title suggests an outline of the associated document topic, it is endowed with a semantic contents that has three functions: Interest and captivate the reader, inform the reader, introduce the topic of the text.

A title is not exactly the smallest possible abstract. While a summary, the most condensed form of a text, has to give an outline of the text contents that respects the text structure, a title indicates the treated subject in the text without revealing all the content [START_REF] Wang | Multi-document summarization using sentencebased topic models[END_REF]. Summarization might rely on titles, such as in [START_REF] Goldsteiny | Summarizing text documents: Sentence selection and evaluation metrics[END_REF] where titles are systematically used to create the summary. This method stresses out the title role, but also the necessity to know the title to obtain a good summary. Text compression could be interesting for titling if a strong compression could be undertaken, resulting in a single relevant word group. Compression texts methods (e.g. [START_REF] Yousfi-Monod | Sentence compression as a step in summarization or an alternative path in text shortening[END_REF]) could be used to choose a word group obeying to titles constraints. However, one has to largely prune compression results to select the relevant group [START_REF] Teufel | Sentence extraction and rhetorical classification for flexible abstracts[END_REF].

A title is not an index : A title does not necessarily contain key words (and indexes are key words), and might present a partial or total reformulation of the text (what an index is not).

Finally, a title is a full entity, has its own functions, and titling has to be sharply distinguished from summarizing and indexing.

It was noticed that elements appearing in the title are often present in the body of the text [START_REF] Zajic | Automatic headline generation for newspaper stories[END_REF]. [START_REF] Baxendale | Man-made index for technical literature -an experiment[END_REF] has showed that the first and last sentences of paragraphs are considered important. The recent work of [START_REF] Belhaoues | Titrage automatique de pages web[END_REF] [START_REF] Jacques | Titres et structuration des documents[END_REF] [START_REF] Zhou | Headline summarization at isi[END_REF] supports this idea and shows that the covering rate of those words present in titles, is very high in the first sentences of a text. [START_REF] Vinet | L'aspet et la copule vide dans la grammaire des titres[END_REF] notices that very often, a definition is given in the first sentences following the title, especially in informative or academic texts, meaning that relevant words tend to appear in the beginning since definitions introduce the text subject while exhibiting its complex terms. The latter indicate relevant semantic entities and constitute a better representation of the semantic document contents [START_REF] Mitra | An analysis of statistical and syntactic phrases[END_REF].

Therefore, this article will first describe a statis-tical analysis of the corpus titles, for each category (e.g., coverage rate, words number, presence of common nouns, verbs, and so forth). The provided corpus is a bunch of articles which have been titled by their authors. The specific features are studied in order to shape a titling process methodology, mostly relying on statistics and lexical selection.

Coverage Rate of Titles Words

To analyze the behavior of human-based titles and subtitles, a corpus of journalistic articles, using the Factiva database (http://factiva.com/), was built. It lists, among others, newspapers articles. The studied corpus contains articles stemming from three French newspapers: Le Monde, Le Figaro, Les Echos. This choice was dependent on the presence of subtitles in articles. The corpus contains 300 articles, that is, 300 titles, covering varied domains (politics, sport, society, sciences). Subtitles are about 354. The corpus admits a total of 169, 796 words. We were interested in the coverage rate of titles and subtitle words. The coverage rate is based on the presence, and frequency, of a title word within the titled text. In this calculation, functional words were not taken into account (i.e. determiners, prepositions,...), nor was punctuation. These statistics were obtained after texts and titles tagging with TreeTagger [START_REF] Schmid | Probabilistic part-of-speech tagging using decision trees[END_REF], where the basic named entities are tagged with the proper nouns label (NAM in TreeTagger). The results indicate that in our corpus, 66 % of the words contained in the titles are present in the text (idem for subtitles). For titles and subtitles, the coverage rate strictly decreases the further the text is processed, with an exception concerning the last part of the text that increases slightly (See Figure 1 and2). We can thus consider that, at least for those journalistic articles in our corpus, the relevant terms for the titling and subtitling are present at the beginning of the text. Besides, statistics have also pointed out a heavy presence of common nouns and named entities with regard to verbs. Therefore, the main idea is to determine the most relevant noun phrase of the text, and use it as title. Thus, the method first stage consisted in extracting a set of candidate noun phrases for titling.

The Automatic Titling Approach

The automatic titling process of a given set of textual data, is performed in two stages presented in

Determining (best) Title(s)

Since a title has to be representative and informative of the text contents, a basic intuitive line leads to select the most "frequent" NP in the text, with a sensible definition of frequency. For that, using TF-IDF [START_REF] Salton | Term-weighting approaches in automatic text retrieval[END_REF] to compute the score of every extracted noun phrase from the text, and then ranking NPs according to this score, has seemed to be a reasonable way of implementing the representativity requirement. However, if a new article is inserted into the corpus, TF-IDF has to be computed again. A first score, N P T F -IDF is computed for each NP. It is the sum of each term TF-IDF, present in the NP (except functional words) [1].

N P T F -IDF = n term=1 (T F * IDF ) term (1)
The main inconvenience of this score is that it does not take into account the NP position in the text, thus neglecting a precious information provided by literature as well as the data statistical analysis (sections 2 and 3). So, if two noun phrases, N P 1, found at the beginning of a text and N P 2 , anywhere in the middle, obtain an identical score, they will be considered as having the same degree of relevance, which disagrees with the idea that first sentences (and sometimes the last ones) are the most promising areas to mine for relevant titles. Thus, this score is corrected by considering the NP position information in the text (N P P OS ). The statistical study showed that the presence of the words of human-defined titles decreases the further the text is processed [START_REF] Zajic | Automatic headline generation for newspaper stories[END_REF], except for the end of the text where it regains some of its previous importance. So the method incorporates a position score N P P OS . It takes into account the position of the NP in the text. Computing it goes as following: The text is divided into several segments of equal sizes (considering the number of words). n is the number of segments of the text and P is the part of the text where appear the noun phrases (P ∈ [1, n]). Since the same study showed that the maximal coverage rate (CR) is obtained at the beginning of the text, then the score needs to decrease in the same proportion. Furthermore, CR decreases abruptly in the first two parts of the text, then moderately until last but one part. This phenomenon is well formalized with an exponential function (see Figure 3)[2]

N P P OS (P ) =    e 1-P if P ∈ [1, n -2] e 2-n if P = n -1 e 3-n if P = n
(2) Finally, N P P OS [2] formula faithfully translates the global aspect of the coverage rate, which weakens until n-2 and modestly grows from n-2 on. Locally, this function offers a hyperbolic curve centered around n -2 1 . The information about the NP position is translated by the score N P P OS that enables to correct the score computed by the 1 for which N PP OS (n -3) = N PP OS (n -1) and N PP OS (n -4) = N PP OS (n) TF-IDF (N P T F -IDF ). The coefficient λ variation balances the position score as well as the T-F.IDF score -[3]. The optimal value of λ ∈ [0, 1] for our corpus is discussed in the section 5.1.

N P score (P ) = λ×N P P OS + (1-λ)×N P T F -IDF

(3)

Evaluation

The purpose of the evaluation presented in this section, is double. First, the 'on-surface evaluation' consists in estimating the automatically determined candidate titles relevance on a set of various texts. It can be associated with a 'deep evaluation' tackling the choice of the 'best' NP(s) among all the extracted NPs. The conclusion of these evaluations points at an optimal value for λ. In this study, we define n = 8, i.e.,each text is segmented in 8 parts of identical size. This figure has been empirically obtained from corpora features (manual) observation.

On-surface Evaluation

The first evaluation is performed on 90 French journalistic articles extracted from our corpus (30 articles of each of the three presented newspapers). Articles retained for this evaluation are the thirty first ones published (from September 11th to September 15th 2010) in Le Monde, Les Echos, and Le Figaro, with the requirement that they present at least one subtitle. The variation of λ between 0 and 1 determines the value adapted to the corpus. All in all, 270 titles were manually estimated (30 articles, so 30 titles according to 9 values for λ). For each title, an expert attributed one of the two following labels, "relevant title" or "irrelevant title". Many candidates for representing a title are acceptable. A relevant title is a well formed word group giving a relevant outline of the text contents. The results indicate that for λ = 0, 25 articles were titled in a relevant way, against only 8 for λ = 1. The best results of automatic titling are obtained with 0.4 ≤ λ ≤ 0.6. It thus seems that, for the given corpus, relevance (i.e. N P T F -IDF ) and position (i.e. N P P OS ) are equally important. So, by defining λ = 0.5, our method attributes a relevant title to two articles over three (58 relevant titles for 90 articles). Several titles (thus several NPs) could be relevant for the same article. So, it is necessary to study the relevance of the chosen NPs among all the extracted NPs.

In-depth evaluation

This evaluation has been performed on three journalistic articles (one from each newspaper), amounting 1, 681 words. All extracted NPs were manually estimated. Many candidates can be juged as relevant for a same article. The evaluating protocol rationale is more to get a fine grained appraisal, than to have a quantitative score. -measure 11.32 11.32 11.32 11.32 11.32 11.32 11.32 11.32 11.32 11.32 11.32 Table 3: Evaluation of journalistic titles (%). T: Threshold.

Conclusion

In this paper, we have tried to sketch a method that automatically extracts and ranks noun phrases (NPs) from untitled texts, to be used as possible titles. Titling web pages and texts has appeared to be a requirement for Web content accessibility, thus pushing researchers to contemplate this task as a useful tool for users. Headlines, or titles, are required to be much shorter than most 'summaries', as well as syntactically well-formed (which disqualified pure lexical approaches) and semantically representative, thus needing a frequency measure. This has led us to choose small syntactic patterns for candidate titles, and corpus observation has highlighted the role of NPs as a good choice. Choosing the most relevant NP for the role of a headline, or at least ranking NPs according to criteria accounting for that relevance, determined the importance of two particular items: The NP position in the text, and the TF-IDF score of its meaningful components. They helped extracting relevant NPs for titling, among all the NPs extracted by syntactical patterns. Evaluation has shown that relevant titles were provided for French journalistic articles with a satisfactory estimation. Among the pending questions, two appear as the most urgent to tackle: First, has the corpus style (e.g. journalistic, scientific , e-commerce or information web sites...) an influence on the method? On which particular criteria does it impact the method: Nature of the patterns; Value of the λ coefficient; Modification of the threshold value? Those are possible tracks to deal with. The second most urgent deals with the first of these, e.g., in addressing verb phrases within the syntactical patterns, and extracting new types or possibly longer titles (as it happens in scientific articles). Further, automatic generation could be contemplated for titling, to produce titles with reformulation or metaphoric features.

Figure 1 :

 1 Figure 1: Curves presenting the distribution of title words in the text.

Figure 2 :

 2 Figure 2: Curves presenting the distribution of subtitle words in the text.

Figure 3 :

 3 Figure 3: Function N P P OS (P )

Table 1 :

 1 Features of journalistic titles

	Newspapers	Le Monde Le Figaro Les Echos Average
	Length of titles (avg.)	6.3	4.5	5.5	5.3
	Verbs (%)	55	52	68	58
	Common Nouns (%)	99	98	99	99
	Nammed Entities (%)	75	70	72	72
	Covery Rate (%)	66	65	68	66
	Newspapers	Le Monde Le Figaro Les Echos Average
	Length of titles (avg.)	2.7	2.5	2.4	2.5
	Verbs (%)	5	7	10	8
	Common Nouns (%)	99	98	100	99
	Nammed Entities (%)	7	16	12	12
	Covering Rate (%)	55	82	74	70

Table 2 :

 2 Features of journalistic subtitles

Table 3

 3

	presents the in-depth evaluation values for preci-
	sion, recall, and F-measure with λ ∈ [0, 1]. The
	threshold, between 5% and 40% (beyond 40%,
	the results are similar), corresponds to the num-
	ber of NPs found by the automatic method, with
	regard to the total number of NP extracted by the
	proposed syntactical filters. It is interesting to
	study the presence of relevant titles found by our
	method according to the threshold, knowing that
	several relevant titles can appear in the list of NPs.
	For instance, if 260 NPs are extracted from the
	text, a threshold of 10% indicates that the best 26
	NPs (with the highest N P Score ) extracted by our
	method, are proposed to the user. A good qual-
	ity system will propose the best relevant titles at
	the top of the classification. The results in Table
	1 indicate that the most relevant titles are obtained
	for 0.30 ≤ λ ≤ 0.90 (F-measure = 59,74%) with
	a threshold of 5%. Finally, the most relevant ti-
	tles are among the first NPs, ranked by (N P score ),
	from the highest to the lowest. Let us notice that
	with λ between 0.30 and 0.90, the recall reaches
	100% with a threshold of 10%. In other words,
	in a more general way, our method gathers all the
	relevant NPs to serve as titles, at the top of its clas-
	sification.
	T	λ	0	0.10	0.20	0.30	0.40	0.50	0.60	0.70	0.80	0.90	1
	5%	Precision	3	22	28	44	44	44	44	44	44	44	15
		Recall	0	56	76	93	93	93	93	93	93	93	17
		F-measure	0	31.59 40.92 59.74 59.74 59.74 59.74 59.74 59.74 59.74 15.94
	10% Precision	4	17	21	24	24	24	24	24	24	24	21
		Recall	20	93	93	100	100	100	100	100	100	100	83
		F-measure 6.67 28.75 34.26 38.71 38.71 38.71 38.71 38.71 38.71 38.71 33.52
	20% Precision	12	13	12	12	12	12	12	12	12	12	11
		Recall	100	100	100	100	100	100	100	100	100	100	100
		F-measure 21.43 23.01 21.43 21.43 21.43 21.43 21.43 21.43 21.43 21.43 19.82
	40% Precision	6	6	6	6	6	6	6	6	6	6	6
		Recall	100	100	100	100	100	100	100	100	100	100	100
		F