Yves Martinez-Maure
email: yves.martinez-maure@sorbonne-universite.fr

On the concurrent normals conjecture for convex bodies

Keywords: 52A20, 52A30, 52A40 57R70 ; Convex bodies, hedgehogs, focal hypersurfaces, concurrent normals conjecture

It is conjectured that any convex body in R n has an interior point lying on normals through 2n distinct boundary points. This concurrent normals conjecture has been proved for n = 2 and n = 3 by E. Heil. J. Pardon put forward a proof for n = 4. For n 5, it is only known that any convex body in R n has an interior point lying on normals through six distinct boundary points. For n 2 f3; 4g, we prove that any normal through a boundary point to any convex body K (with a smooth enough support function) in R n passes arbitrarily close to the set of interior points of K [ L lying on normals through at least 6 distinct points of @K, where L is the body bounded by the smallest convex parallel hypersurface to @K whose unit normal points in the opposite direction. This study leads us to introduce and study new concepts for studying focals of closed convex hypersurfaces in R n+1 . Finally, we prove that for some convex body K of R 4 , there are only 6 normal lines passing through the center of the minimal spherical shell. 0 2020 MSC:

Introduction and statements of main results

It is conjectured that any convex body in n-dimensional Euclidean space R n has an interior point lying on normals through 2n distinct boundary points. Using the existence of a minimal spherical shell for any convex body and, a combination of Morse theory and approximation, E. Heil has proved this concurrent normals conjecture for n = 2 and n = 3 in [START_REF] Heil | Existenz eines 6-Normalenpunktes in einem konvexen Körper[END_REF][START_REF] Heil | Correction to 'Existenz eines 6-Normalenpunktes in einem konvexen Körper[END_REF][START_REF] Heil | Concurrent Normals and Critical Points under Weak Smoothness Assumptions[END_REF]. For n = 4, J. Pardon put forward a proof of the conjecture under a smoothness assumption on the boundary [START_REF] Pardon | Concurrent normals to convex bodies and spaces of Morse functions[END_REF]. For n 5, it is only known that any convex body in R n has an interior point lying on normals through six distinct boundary points. However Zam…rescu has shown that, in the sense of Baire category based on the Hausdor¤ distance between convex sets, most interior points of most convex bodies lie on in…nitely many normals [START_REF] Zamfirescu | Points on in…nitely many normals to convex surfaces[END_REF]. For n 2 f3; 4g, we prove that any normal through a boundary point to any convex body K (with a smooth enough support function) in R n passes arbitrarily close to the set of interior points of K [ L lying on normals through at least 6 distinct points of @K, where L is the body bounded by the smallest convex parallel hypersurface to @K whose unit normal points in the opposite direction.

Our setting and tools

In this paper, we assume for the sake of simplicity of the presentation that the support function is C 1 but our results remain true provided that the support function is smooth enough (say at least of class C 4 ). Actually, our main arguments essentially relies on the Morse lemma, which remains true for C 2 -functions [START_REF] Ostrowski | On the Morse-Kuiper theorem[END_REF], and on elementary properties of the focal of the boundary.

In most of the papers on concurrent normals to a convex body K with a smooth boundary @K in R n+1 , the focal (or evolute) of @K is regarded as the complement of the set of points x 2 R n+1 such that the square of the distance function from x induces a Morse function on @K. In this paper, we will adopt another point of view. For any x 2 R n+1 , we will consider the support function of @K with respect to x, that is h x : S n ! R, u 7 ! h (u) hx; ui, where h : S n ! R is the support function of K, and we will regard the evolute of @K as the complement of set of points x 2 R n+1 such that h x : S n ! R is a Morse function.

We will also make intensive use of the so-called 'hedgehogs'of R n+1 , which are the (possibly singular and self-intersecting) hypersurfaces of R n+1 that are parametrized by their Gauss map and parallel to some C 2 convex hypersurface in R n+1 . Every h 2 C 2 (S n ; R) can be regarded as the support function of the hedgehog (hypersurface) H h of R n+1 that is parametrized by the map x h : S n ! H h R n+1 , u 7 ! h(u)u + (rh) (u), which may be interpreted as the inverse of the Gauss map, in the sense that: at each regular point x h (u) of H h , u is a normal vector to H h (see Section 2). Since all the hedgehogs that are parallel to a same C 2 convex hypersurface with support function h 2 C 2 (S n ; R) have the same evolute and support functions that only di¤er by a constant, we will denote this common evolute by F rh , and place ourselves in the setting of hedgehogs.

For h 2 C 1 (S n ; R), and u 2 S n , the normal line to the hedgehog H h at x h (u) is de…ned to be the line passing through x h (u) and oriented by u; this normal line N rh (u) := frh (u)g + Ru is the perpendicular to the support hyperplane to H h at x h (u). The focal set (or evolute) F rh of H h can be de…ned as the locus of its centers of principal curvatures, or equivalently, as the envelope of its normal lines (N rh (u)) u2S n .

Our main statements

For n = 2, we will prove the following result which will turn out to be a re…nement of Heil's theorem.

Theorem 1. Let H h be a hedgehog of R 3 such that h 2 C 1 (S 2 ; R), and let F rh denote its focal surface. If u 2 S 2 is such that the normal line N rh (u) to H h at x h (u) does not meet the singular locus of F rh , then there exists some r 2 R such that x h r (u) = x h (u) ru 2 N rh (u) is an (at least) double hyperbolic point of H h r .

Here, "x h r (u) = x h (u) ru 2 N rh (u) is an (at least) double hyperbolic point of H h r "means that there exists v 2 S 2 nfug, such that:

x h r (u) = x h r (v) , R h r (u) < 0, and R h r (v) < 0 where R h r is the curvature function of H h r (that is, the inverse 1= h r of the Gauss curvature h r of H h r ).

We will deduce the following reformulation of Heil's theorem without making use of the notion of a minimal spherical shell.

Corollary 1 (Heil' s theorem [START_REF] Heil | Existenz eines 6-Normalenpunktes in einem konvexen Körper[END_REF][START_REF] Heil | Correction to 'Existenz eines 6-Normalenpunktes in einem konvexen Körper[END_REF]). Let H h be a hedgehog of R 3 such that h 2 C 1 (S 2 ; R). Either there exists a point of R 3 lying on in…nitely many normals to H h or there exists an open set formed by points of R 3 n F rh lying on at least 6 normals to H h .

We will in fact prove the following stronger result.

Corollary 2 Let H h be a hedgehog of R 3 such that h 2 C 1 (S 2 ; R). If there does not exist a point of R 3 lying on in…nitely many normals to H h , then in…nitely many normals to H h do not meet the singular locus of F rh , and any one of them meets the closure of x 2 R 3 n F rh jN rh (x) 6 , where N rh (x) denotes the number of normal lines to H h passing through x.

The focal (or evolute) F rh of H h is the singular hypersurface of R n+1 formed by all the principal centers of curvature of H h and it consists of n sheets F 1 rh ; : : : ; F n rh corresponding respectively to the principal radii of curvature R 1 h ; : : : ; R n h of H h , which we label so that R 1 h R 2 h : : : R n h . In Subsection 2.3, we will introduce for each k 2 [ j1; nj ] the index of a point x 2 R n+1 nF rh with respect to the k th sheet F k rh of F rh , and the interior Int F k rh of this sheet. Under the assumptions of Corollary 2, we will in fact prove that any normal to H h that does not meet the singular locus of F rh meets the closure of the nonempty interiors of both sheets of the focal.

We will obtain the following result as a corollary of our Theorem 3 stated and proved in Subsect 4.2.

Theorem 2. Let H h be a hedgehog of R 4 such that h 2 C 1 (S 3 ; R), let F rh denote its focal surface, and let F 2 rh be its second sheet. For all x 2 R 4 n F rh , we have:

x 2 Int F 2 rh ()
h x : S 3 ! R admits a smooth level surface with nonzero genus :

The theorem below is an adaptation of Theorem 1 to dimension 4.

Theorem 4. Let H h be a hedgehog of R 4 such that h 2 C 1 (S 3 ; R), and let F rh denote its focal surface. If u 2 S 3 is such that the normal line N rh (u) to H h at x h (u) does not meet the singular locus of F rh , then there exists (r

1 ; r 2 ) 2 R 2 such that, for each i 2 [ j1; 2j ], x h r i (u) = x h (u) r i u is an (at least) type i double hyperbolic point of H h r i . Corollary 3 Let H h be a hedgehog of R 4 such that h 2 C 1 (S 3 ; R).
If there does not exist a point of R 4 lying on in…nitely many normals to H h , then in…nitely many normals to H h do not meet the singular locus of F rh and any one of them meets the closure of

x 2 R 4 n F rh jN rh (x) 6 ,
where N rh (x) denotes the number of normal lines to H h passing through x; more precisely, any one of these normals meets the closures of

Int F 1 rh \ Int F 2 rh and Int F 2 rh \ Int F 3 rh .
Finally, we will prove that it is not true that for any convex body K of R 4 , there are at least 8 normal lines passing through the center of the minimal spherical shell of K (Theorem 6).

2 Basics on hedgehogs and their evolutes

Background on hedgehogs

Classical hedgehogs can be regarded as the geometrical realizations of formal di¤erences of convex bodies in the Euclidean vector space R n+1 . The idea of considering the Minkowski di¤erences of convex bodies may be traced back to some papers by A.D. Alexandrov [START_REF] Aleksandrov | Zur Theorie der gemischten Volumina von konvexen Körpern, I:Verallgemeinerung einiger Begri¤e der Theorie der konvexen Körper (in Russian)[END_REF] and H. Geppert [START_REF] Geppert | Über den Brunn-Minkowskischen Satz[END_REF] in the 1930's. H. Geppert introduced hedgehogs for n 2 under the German names stützbare Bereiche (n = 1) and stützbare Flächen (n = 2). Many classical notions for convex bodies extend to hedgehogs and quite a number of classical results …nd their counterparts. Of course, a few adaptations are necessary. In particular, volumes have to be replaced by their algebraic versions. Hedgehogs have proved useful for studying convex bodies (one of the main successes of the theory is the construction of counterexamples to an old conjectured characterization of the 2-sphere [START_REF] Martinez-Maure | Contre-exemple à une caractérisation conjecturée de la sphère[END_REF][START_REF] Martinez-Maure | New notion of index for hedgehogs of R 3 and applications[END_REF][START_REF] Panina | New counterexamples to A. D. Alexandrov's hypothesis[END_REF]), and for geometrizing analytical problems by considering functions as support functions. This section will provide the reader with the necessary background on hedgehogs in order to facilitate the understanding of the following sections.

The C 2 case

Here we follow more or less [START_REF] Langevin | Hérissons et multihérissons (enveloppes paramétrées par leur application de Gauss)[END_REF]. 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 de…ned by h K (u) = sup fhx; ui jx 2 K g, (u 2 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 Gauss 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 hx; ui = h(u). This envelope H h is described analytically by the following system of equations hx; ui = h(u) hx; : i = dh u (:) .

The second equation is obtained from the …rst by performing a partial di¤erentiation with respect to u. From the …rst 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 (see Figure 1). Therefore, for each u 2 S n , x h (u) = h(u)u + (rh) (u) is the unique solution of this system.

Figure 1. Envelope parametrized by its Gauss map Now, for any C 2 -function h on S n , the envelope H h is in fact well-de…ned (even if h is not the support function of a convex hypersurface). Its natural parametrization x h : S n ! H h ; u 7 ! h(u)u + (rh) (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 (see Figure 2). Note that x h depends linearly on h.

Since x h : S n ! H h can be regarded as the inverse of the Gauss map, the Gauss curvature 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. Singular points of H h are the points at which the the so-called 'curvature function'R h (u) := det [T u x h ] is equal to 0 (that is, loosely speaking, the points at which the Gauss curvature h is in…nite). For every u 2 S n , the tangent map of x h at the point u is T u x h = h(u) Id TuS n + L u (r 2 h), where L u (r 2 h) is the symmetric endomorphism associated with the Hessian (r 2 h) u of h at u. In particular, the so-called 'curvature function' + such that h = (h + r) r. By regarding hedgehogs as 'Minkowski di¤erences of arbitrary convex bodies', the notion of a hedgehog of R n+1 can be extended to hedgehogs whose support functions belong to the linear subspace of C (S n ; R) that is spanned by all the support functions of convex bodies of R n+1 (see [START_REF] Martinez-Maure | Geometric study of Minkowski di¤erences of plane convex bodies[END_REF]).

R h (u) := det [T u x h ] is given by R h (u) = det [h(u) Id TuS n + L u (r 2 h)] for all u 2 S n .

Generic singularities

Hedgehogs of R n+1 with a smooth support function can be regarded as Legendrian fronts [14, pp. 340-341], and by regarding the natural parametrizations x h : S n ! R as Legendrian maps, Arnold's works can be used to classify their generic singularities for n 5 [START_REF] Langevin | Hérissons et multihérissons (enveloppes paramétrées par leur application de Gauss)[END_REF]. In particular, generic singularities of smooth hedgehogs of R 3 are cuspidal edges and swallowtails. Elliptic and hyperbolic regions, which are de…ned by the sign of the Gauss curvature h = 1=R h , are separated by cuspidal edges on which the curvature function R h is equal to 0 (or, loosely speaking, on which the Gauss curvature h is in…nite): see Figure 3 (a). Swallowtails are the cusp points of cuspidal edges and we can distinguish two types of swallowtails (negative or positive) according to the sign of the Gauss curvature on the tail: see Figure 3 (a) and (b). More precisely, there exists an open dense subset U of C 1 (S 2 ; R) in the C 4 -topology, such that: for all h 2 U, the singularities of H h are all equivalent to one of the three models of singularities represented in Figure 3.

Focal (or evolute) of a hedgehog H h R n+1

Let H h be a hedgehog of R n+1 such that h 2 C 1 (S n ; R). For all u 2 S n , the normal line to H h at x h (u) is de…ned to be the line passing through x h (u) and oriented by u; this normal line N rh (u) := frh (u)g + Ru is the perpendicular to the support hyperplane to H h at x h (u). The focal set (or evolute) of H h is the locus of its centers of principal curvatures, or equivalently, the envelope of its normal lines. This set is also the union of all the singular points of all the parallel hedgehogs H h+r , (r 2 R). Since it only depends on the gradient of h, we will denote it by F rh . The focal (or evolute) F rh of H h is the singular hypersurface of R n+1 formed by all the principal centers of curvature of H h and it consists of n sheets F 1 rh ; : : : ; F n rh corresponding respectively to the principal radii of curvature R 1 h ; : : : ; R n h of H h , which we label so that R

: S n ! R n+1 ; u 7 ! c k rh (u) = x h (u) R k h (u) u = rh (u) k rh (u) u, since x h (u) = rh (u)+h (u) u and R i h (u) = k rh + h (u)
, where rh (u) is the gradient of h at u, and 1 rh (u) ; : : : ; n rh (u) are the respective eigenvalues of the Hessian of H h at u.

In most of the papers on concurrent normals to a convex body K with a smooth boundary @K in R n+1 , the evolute of @K is also regarded as the complement of the set of points x 2 R n+1 such that the square of the distance function from x induces a Morse function on @K:

d x : @K ! R y 7 ! kx yk 2 ,
where k:k : R n+1 ! R + is the Euclidean norm.

In this paper, we will adopt another point of view. For any x 2 R n+1 , we will consider the support function of H h with respect to x, that is h x : S n ! R, u 7 ! h (u) hx; ui, and we will regard the evolute F rh as the complement of set of points x 2 R n+1 such that h x : S n ! R is a Morse function. In other words, we will regard the evolute of H h as the subset F rh of R n+1 on which the number and nature of the critical points of h x change. We will make use of singularity theory and Morse theory viewpoints in order to describe and summarize these changes. In particular, we will make essential use of the following proposition.

Proposition 1. Let H h be a hedgehog of R n+1 such that h 2 C 1 (S n ; R), and let x 2 R n+1 n F rh . For all u 2 S n , the normal line N rh (u) to H h at x h (u) is passing through x if, and only if, r (h x ) (u) = 0. Thus, the number of normal lines to H h passing through x is given by

N rh (x) = # fu 2 S n jr (h x ) (u) = 0 g
Proof. The point fxg can be regarded as the hedgehog with support function l x : S n ! R; u 7 ! hx; ui, and for all u 2 S n , we have:

x = l x (u) u + r (l x ) (u) = hx; ui u + u ? (x) ,
where u ? (x) is the orthogonal projection of x onto the linear hyperplane that is orthogonal to u. Therefore, the hedgehog H hx is the Minkowski di¤erence H h fxg, and for all u 2 S n , we have:

x hx (u) = h x (u) u+r (h x ) (u) = x h (u) x = h x (u) u+(rh (u) u ? (x)) , so that r (h x ) (u) = rh (u) u ? (x)
. Now, for all u 2 S n , we thus have:

(x 2 N rh (u)) , (9r 2 R; x h (u) + ru = x) , (9r 2 R; (h x (u) + r) u + rh (u) u ? (x) = 0) , (rh (u) u ? (x) = 0) , (r (h x ) (u) = 0) :

Singularity and Morse theories viewpoints

In this subsection, we essentially follow [18, 9.1]. Given h 2 C 1 (S n ; R), we consider the following family of functions

F h : U R n+1 := R n+1 S n ! R (x; u) 7 ! h x (u) = h (u) hx; ui :
The critical set (or catastrophe manifold) of F h is the set

C rh = (x; u) 2 U R n+1 @F h @u (x; u) = 0 i:e: r (h x ) (u) = 0 .
The catastrophe map := rh is the restriction to the critical set C rh of the projection : U R n+1 ! R n+1 , (x; u) 7 ! x. The singularity set S rh is the set of points of C rh at which the catastrophe map has rank less than n + 1:

S rh := (x; u) 2 C rh rk T (x;u) < n + 1 = (x; u) 2 C rh 9k 2 [ j1; nj ] ; x = c k rh (u) = f(x; u) 2 C rh j9k 2 [ j1; nj ] ; 0 = c k rh (u) x = c k r(hx) (u) = r (h x ) (u) | {z } =0 k r(hx) (u)u 9 > = > ; = (x; u) 2 C rh 9k 2 [ j1; nj ] ; k r(hx) (u) = 0 = (x; u) 2 C rh det r 2 (h x ) (u) = 0 ,
where r 2 is the Hessian operator and thus M = det r 2 the Monge-Ampère one. It is thus the set of points (x; u) 2 C rh at which F h : U R n+1 ! R has a degenerate critical point. The bifurcation set of F h : U R n+1 ! R is de…ned to be the image of the singularity set S rh under the catastrophe map . It is none other than the focal F rh of the hedgehog H h in R n+1 (and of any hedgehog that is parallel to H h , that is of the form H h+ , where 2 R):

(S rh ) = F rh = n [ k=1 F k rh ,
where

F k rh := c k rh (S n ) = x 2 R n+1 9u 2 S n , x = c k rh (u) = x 2 F rh 9u 2 S n , r (h x ) (u) = 0 and k r(hx) (u) = 0 . Note that R n+1 n F rh = x 2 R n+1 jh x : S n ! R is a Morse function ,
while the bifurcation set F rh = (S rh ) is the subset of R n+1 on which the number and nature of the critical points of h x : S n ! R change (for by structural stability of Morse functions such a change can only occur passing through a degenerate critical point). This can be of course be checked by a direct computation.

Index of a point with respect to the focal

Of course, every x belonging to the unbounded connected component of R n+1 n F rh lies on exactly two normal lines to H h . We have proved in Proposition 1 that, for every x 2 R n+1 n F rh , the number of normal lines to H h passing through x is given by: Theorem 4], and it suddenly changes by two units every time x transversally crosses F rh at a simple regular point, so that N rh (x) is even for any x 2 R n+1 n F rh . It is thus natural to de…ne, for every x 2 R n+1 n F rh , the index i F rh (x) of x with respect to F rh by putting:

N rh (x) = # fu 2 S n jr (h x ) (u) = 0 g . This number N rh (x) is constant and even on any connected component of R n+1 n F rh [7,
i F rh (x) := 1 1 2 N rh (x) ,
This index induces a transverse orientation of the regular part of F rh (and thus of any of its n sheets): F rh is transversely oriented so that the number of normal lines to H h passing through x increases by two units when x transversally crosses F rh at a simple regular point in the direction of the transverse orientation. Let x 2 R n+1 n F rh so that h x : S n ! R is a Morse function. Denote by C m (x) and C M (x) the respective numbers of critical points of index 0 and n of this Morse function (that is, the respective numbers of its local minima and maxima), and, for every i 2 [ j1; n 1j ], denote by S i (x) the number of its critical points of index i. We know that the number of normal lines to H h passing through x is then given by:

N rh (x) = # (fu 2 S n jr (h x ) (u) = 0 g) = C m (x) + n 1 X i=1 S i (x) + C M (x) : (1)
Furthermore, by virtue of the Morse-Euler relationship, we have:

C m (x) + n 1 X i=1 ( 1) i S i (x) + ( 1) n C M (x) = (S n ) , (2)
where (S n ) is the Euler characteristic of S n , i.e. 1 + ( 1) n . From (1) and (2), we can immediately deduce that:

i F rh (x) := 8 > > > > > > > > > > < > > > > > > > > > > : p X l=1 S 2l 1 (x) if n = 2p 2 2Z 1 C m + p X l=1 S 2l ! (x)
, or equivalently,

1 C M + p 1 X l=0 S 2l+1 ! (x) if n 1 = 2p 2 2Z , (3)

Index decomposition

For any k 2 [ j1; nj ], assume that x moves and transversally crosses the k th sheet F k rh of F rh at a simple regular point of F k rh in the direction of the transverse orientation. We know this crossing of F k rh results in a twounits increase in the number of normal lines to H h passing through x (i.e. in the number of critical points of h x : S n ! R). From the Morse-Euler relationship, the two new critical points of h x cannot be of the same index (since the two indices must have di¤erent parities). Since, moreover, F k rh is given by

F k rh = x 2 F rh 9u 2 S n , r (h x ) (u) = 0 and k r(hx) (u) = 0 ,
it appears that the two critical points of h x : S n ! R that arise at the moment of the crossing of F k rh have adjacent indices k 1 and k, the sign of the function k r(hx) : S n ! R being di¤erent at these two points. Therefore, this crossing of F k rh has the e¤ect of transforming the vector v (x) = (C m (x) ; S 1 (x) ; : : : ; S n 1 (x) ; C M (x)) into the vector v (x) + e k 1 + e k , where (e 0 ; : : : ; e n ) is the canonical basis of R n+1 . Now, let us distinguish two cases according to the parity of n 1. In both cases, and for every x 2 R n+1 n F rh , we will split the index i F rh (x) of x with respect to the focal

F rh into the sum over k 2 [ j1; nj ] of the (appropriately de…ned) index i F k rh (x) of x with respect to the k th sheet F k rh of F rh . In the case that n = 2p 2 2Z , de…ne the indices i F k rh (x); (k 2 [ j1; 2pj ]), by 8 > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > : i F 1 rh (x) = 1 C m (x) i F 2 rh (x) = (C m S 1 ) (x) 1 . . . i F p rh (x) = ( 1) p 1 1 C m S 1 + : : : + ( 1) p 1 S p 1 (x) i F p+1 rh (x) = ( 1) p 1 1 C M S 2p 1 + : : : + ( 1) p 1 S p+1 (x) . . . i F 2p 1 rh (x) = (C M S 2p 1 ) (x) 1 i F 2p rh (x) = 1 C M (x)
for all x 2 R 2p+1 n F rh , so that when x moves and transversally crosses the k th sheet F k rh of F rh at a simple regular point of F k rh in the direction of the transverse orientation, then the index i F k rh (x) decreases by one unit. Thus, for every x 2 R 2p+1 n F rh , i F k rh (x) can be interpreted as the index of x with respect to F k rh equipped with its transverse orientation. Note that, for every x 2 R 2p+1 n F rh , the 2p strong Morse inequalities, which give lower bounds for the number of critical points of each index of the Morse function h x : S 2p ! R in terms of the Betti numbers of S 2p , can simply be rewritten as: 8k 2 [ j1; 2pj ], i F k rh (x) 0. In other words, these 2p indices are nonpositive, and, for every k 2 [ j1; 2pj ], the index i F k rh (x) negatively measures how far from equality we are in the k th strong Morse inequality when considering the Morse function h x : S 2p ! R.

Similarly, in the case that

n 1 = 2p 2 2Z , de…ne the indices i F k rh (x) ; (k 2 [ j1; 2p + 1j ]), by: 8 > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > : i F 1 rh (x) = 1 C m (x) i F 2 rh (x) = (C m S 1 ) (x) 1 . . . i F p+1 rh (x) = ( 1) p 1 C m + p X l=1 ( 1) l S l ! (x) ! . . . i F 2p rh (x) = (C M S 2p 1 ) (x) 1 i F 2p+1 rh (x) = 1 C M (x) :
: for all x 2 R 2p+2 n F rh , so that when x moves and transversally crosses the k th sheet F k rh of F rh at a simple regular point of F k rh in the direction of the transverse orientation, then the index i F k rh (x) decreases by one unit. Thus, for every x 2 R 2p+2 n F rh , i F k rh (x) can be interpreted as the index of x with respect to F k rh equipped with its transverse orientation. Again, for every x 2 R 2p+2 nF rh , the 2p+1 strong Morse inequalities, which give lower bounds for the number of critical points of each index of the Morse function

h x : S 2p+1 ! R
x : S 2p+1 ! R.
In both cases (n even or odd), using the Morse-Euler relationship we can verify that:

Proposition 2. For every x 2 R n+1 n F rh , we have i F rh (x) = n X k=1 i F k rh (x) ,
and, for every k 2 [ j1; nj ], i F k rh (x) can be interpreted as the index of x with respect to the k th sheet F k rh of F rh equipped with its transverse orientation; the index i F k rh (x) negatively measures how far from equality we are in the k th strong Morse inequality when considering the Morse function h

x : S n ! R, (k 2 [ j1; nj ]).
We de…ne the interior and the body of the focal F rh by setting, respectively,

Int (F rh ) := x 2 R n+1 n F rh ji F rh (x) < 0 , and K (F rh ) := F rh [ Int (F rh ) .
Of course, for every k 2 [ j1; nj ] we can also de…ne the interior and the body of the k th sheet F k rh of the focal by setting, respectively,

Int F k rh := n x 2 R n+1 n F k rh i F k rh (x) < 0 o ,
where i F k rh (x) is de…ned as indicated above, and

K F k rh := F k rh [ Int F k rh .
It has been proved that arbitrarily close to the center of the minimal spherical shell of a convex body K R n+1 with support function h 2 C 1 (S n ; R), there exists points

x 2 R n+1 n F rh such that C m (x) 2 and C M (x) 2 [7, Lemma 3], so that i F 1 rh (x) = 1 C m (x) 1 and i F n rh (x) = 1 C M (x) 1.
In other words, we have:

Int F 1 rh \ Int (F n rh ) 6 = ?.

Singular locus of the focal

F rh of H h R n+1 Here H h is a hedgehog of R n+1 such that h 2 C 1 (S n ; R). When two principal radii of curvature R k h ; R l h of H h coincide at a point u 2 S n , then the corresponding sheets F k rh , F l rh of F rh intersect at c k rh (u) = c l rh (u)
, and this point is a singular point of both sheets F k rh , F l rh . When all the principal radii of curvature are pairwise distinct at u 2 S n , the focal F rh is locally the union of n disjoint patches of hypersurfaces, parametrized by the maps c k rh :

S n ! R n+1 , u 7 ! x h (u) R k h (u) u, (k 2 [ j1; nj ]).
Let us examine the regularity of these patches at their point corresponding to u. In the case that the principal radii of curvature are pairwise distinct at u 2 S n , there exists an orthonormal basis of T u S n consisting of eigenvectors v 1 ; : : : ; v n of the tangent map T u x h associated respectively with R 1 h (u) ; : : :

; R n h (u): 8k 2 [ j1; nj ], (T u x h ) (v k ) = R k h (u) v k . For every k 2 [ j1; nj ],
we have then:

@c k rh @v k (u) = (T u x h ) (v k ) @R k h @v k (u) u + R k h (u) v k = @R k h @v k (u) u,
and for every l 2 [ j1; nj ] n fkg,

@c k rh @v l (u) = (T u x h )(v l ) @R k h @v l (u) u + R k h (u) v l = R l h R k h (u) v l @R k h @v l (u) u.
Denote by B the orthonormal system (u; v 1 ; : : : ; b v k ; : : : ; v n ), where the hat over the term v k means that it must be omitted. A straightforward computation shows that

det B " @c k rh @v k ; @c k rh @v 1 ; : : : ; d @c k rh @v k ; : : : ; @c k rh @v n ! (u) # ,
where the hat means again that the corresponding term must be omitted, is equal to

@R k h @v k (u) Y 1 l n l 6 = k R l h R k h (u) .
Thus:

Proposition 3. For every k 2 [ j1; nj ] and every u 2 S n , c k rh (u) is a singular point of F k
rh if, and only if, one of the following two conditions is satis…ed:

R k h (u) is an eigenvalue of multiplicity at least 2 of the tangent map T u x h ; R k h (u) is a simple eigenvalue of T u x h , and @R k h @v k (u) = 0, v k being a unit eigenvector of T u x h associated with R k h (u).
For every k 2 [ j1; nj ], the singular locus of the k th sheet F k rh of F rh is thus given by:

Sing F k rh = c k rh (S k )
, where S k is the set of point u 2 S n that satisfy one of the two conditions of the above proposition; and the singular locus of the focal F rh is of course de…ned by:

Sing (F rh ) := n [ k=1 Sing F k rh .
The three-dimensional case

Corollary. Let H h be a hedgehog of R 3 such that h 2 C 1 (S 2 ; R), and let u 2 S 2 be such that

R 1 h (u) = 0, R 2 h (u) 6 = 0 and @R 1 h @v 1 (u) 6 = 0 resp. R 2 h (u) = 0, R 1 h (u) 6 = 0 and @R 2 h @v 1 (u) 6 = 0 , where (v 1 ; v 2 ) is an orthonormal basis of T u S 2 made of eigenvectors v 1 ; v 2 of T u x h associated respectively with R 1 h (u), R 2 h (u). Then, x h (u) is equal to c 1 rh (u) (resp. c 2 rh (u))
, which is a regular point of F 1 rh (resp. F 2 rh ), and a regular point of a cuspidal edge of H h .

Proof. The …rst part of the corollary is a straightforward consequence of the proposition. Now, let (i; j) = (1; 2) (resp. (i; j) = (2; 1)) so that:

R i h (u) = 0, R j h (u) 6 = 0 and @R i h @v i (u) 6 = 0.
Since R h (u) = 0 and rR h (u) 6 = 0, the level set R h can be parametrized as a regular smooth curve in a neighborhood of u on S 2 .

Let : I ! S 2 , t 7 ! (t) be this regular parametrization of , and let t 0 2 I be such that (t 0 ) = u and 0 (t

0 ) = 1 v 1 + 2 v 2 ,
where

( 1 ; 2 ) 2 R 2 n f(0; 0)g. Since R h
is identically equal to zero on I, we have (R h ) 0 (t 0 ) = hrR h (u) ; 0 (t 0 )i = 0. On the other hand, @R i h @v i (u) := hrR h (u) ; v i i 6 = 0. Therefore 0 (t 0 ) is not colinear to v i , and thus j 6 = 0. As a result

(x h ) 0 (t 0 ) = (T u x h ) ( 0 (t 0 )) = i R i h (u) v i + j R j h (u) v j 6 = 0, that is u = (t 0 ) is a regular point of the cuspidal edge x h ( ).
Note in passing that any cuspidal edge of H h is locally separating a hyperbolic region from an elliptic one.

Volume of the focal

F rh of H h R n+1
Here again H h is a hedgehog of R n+1 such that h 2 C 1 (S n ; R). We can de…ne the (absolute) volume of its focal F rh , and, for any k 2 [ j1; nj ] the one of the k th sheet F k rh of F rh , to be respectively

v (F rh ) := Z R n+1 i F rh (x) dx, and v F k rh := Z R n+1 i F k rh (x) dx,
where the integrals are with respect to the Lebesgue measure on R n+1 . By the index decomposition, we thus have:

v (F rh ) = n X k=1 v F k rh .
In the case where H h R 3 , the volume of F rh has already been introduced by the author in [START_REF] Martinez-Maure | Dérivation des surfaces convexes de R 3 dans l'espace de Lorentz et étude de leurs focales[END_REF], where the following is proved.

Theorem. Let H h be a hedgehog of R 3 such that h 2 C 1 (S 2 ; R). The volume of its focal F rh is given by

v (F rh ) = 1 6 Z S 2 jR 1 R 2 j 3 d ,
where is the spherical Lebesgue measure on S 2 , and R 1 (u), R 2 (u) are the principal radii of curvature of H h at x h (u). Besides, the map

v : H := rh h 2 C 4 S 2 ; R ! R + ; rh 7 ! v (F rh ) 1 3
is a norm on the real vector space of families of parallel hedgehogs with support function of class C 4 in R 3 .

3 The three-dimensional case

Usual index and transverse orientation

Any hedgehog H h of R n+1 is a (possibly singular and self-intersecting) parametrized hypersurface x h : S n ! H h R n+1 that is equipped with the transverse orientation de…ned as follows: at each regular point x h (u) of H h , the usual transverse orientation of H h is given by the normal vector sgn [R h (u)] u, where sgn is the sign function and R h := 1= h the curvature function of H h ( h denoting the Gauss curvature of H h ). The Kronecker index i h (x) of a point x 2 R n+1 n H h with respect to H h can be de…ned as the degree of the map

U (h;x) : S n ! S n ; u 7 ! x h (u) x kx h (u) xk ,
and interpreted as the algebraic intersection number of an oriented halfline with origin x with the hypersurface H h equipped with its usual transverse orientation (number independent of the oriented half-line for an open dense set of directions). The usual transverse orientation and the Kronecker index are thus mutually associated. It is worth noting that if we let e h (u) = h ( u) for all u 2 S n , where h 2 C 2 (S n ; R), then the hedgehogs H h = x h (S n ) and He h = x e h (S n ) are identical as hypersurfaces of R n+1 except that they have opposite transverse orientations when n + 1 is odd. Indeed

xe h ( u) = x h (u) for all u 2 S n , but sgn R e h ( u) ( u) = ( 1) n+1 sgn [R h (u)] u,
and thus

i e h (x) = ( 1) n+1 i h (x) for all x 2 R n+1 n H h :
For n + 1 = 3, the author proved the following result [13, Theorem 1]:

Theorem. Let H h be a hedgehog of R 3 such that h 2 C 2 (S 2 ; R). Then, for all x 2 R 3 n H h , we have:

i h (x) = r + h (x) r h (x) ,
where r h (x) (resp. r + h (x) denotes the number of connected components of S 2 n h 1

x (f0g) on which h x is negative (resp. positive).

Proof. For the convenience of the reader, we recall the main steps of the proof, which can be useful to understand the following subsection. For all x 2 R 3 n H h , we have r (h x ) (u) 6 = 0 whenever h x (u) = 0, (u 2 S 3 ). Therefore, for all x 2 R 3 n H h , the set h 1 x (f0g) consists of a …nite number, say c h (x), of disjoint simple smooth closed spherical curves on which h x changes sign cleanly. Note that c h (x) = r h (x) + r + h (x) 1. Then, the proof relies on the two following lemmas.

Lemma 1. The map x 7 ! i h (x) r + h (x) r h (x) is constant on R 3 nH h . (The …rst step consists in noticing that x 7 ! r h (x), x 7 ! r + h (x) and thus x 7 ! c h (x) are constant on each connected component of R 3 n H h .
The second one, consists in proving that x 7 ! i h (x) r + h (x) r h (x) remains constant as x transversally crosses an elliptic (resp. hyperbolic) region of H h . As x transversally crosses a simple elliptic region of H h at x h (u) from locally convex to locally concave side, we must distinguish two cases: (i) If u is pointing towards the locally concave side, then i h (x) decreases by one unit whereas r h (x) increases by one unit, and r + h (x) remains constant; (ii) If u is pointing towards the locally convex side, then i h (x) and r + h (x) increases by one unit whereas r h (x) remains constant. As x transversally crosses a simple hyperbolic region of H h at x h (u) in the direction of u, which is the unit normal at x h (u) since x h (u) is hyperbolic, then i h (x) decreases by one unit and there are exactly two possibilities: (i) If c h (x) increases by one unit then r h (x) increases by one unit and r + h (x) remains constant; (ii) If c h (x) decreases by one unit then r + h (x) decreases by one unit and r h (x) remains constant). Lemma 2. If kxk is su¢ ciently large, then c h (x) = 1: (This second lemma essentially follows from the fact that x h : S 2 ! H h can be interpreted as the inverse of the Gauss map).

Lemma 2 implies that r h (x) = r + h (x) = 1 when kxk is su¢ ciently large, and thus the theorem follows from Lemma 1.

Since the usual transverse orientation does depend on the orientation of normal lines to x h (S 2 ) = x e h (S 2 ) in R 3 , we will then rename it 'the relative transverse orientation'of H h in R 3 , so as to distinguish it from the one we de…ne below. We refer the reader to [START_REF] Martinez-Maure | New notion of index for hedgehogs of R 3 and applications[END_REF] for more information on this subject.

New index and transverse orientation

In [START_REF] Martinez-Maure | New notion of index for hedgehogs of R 3 and applications[END_REF], the author introduces the following notion of index of a point x 2 R 3 n H h with respect to H h :

j h (x) := 1 c h (x) ,
where c h (x) denotes the number of connected components of (h x ) 1 (f0g) on S 2 , that is the number of closed spherical curves formed by points u 2 S 2 such that x belongs to the support hyperplane of H h at x h (u). From the above, the index j h : x 7 ! j h (x) remains constant on each connected component of R 3 n H h . In particular, j h is equal to 0 on the unbounded component of R 3 n H h . It is worth noting that the value of j h (x) must obviously decreases as x transversally crosses H h at a simple elliptic point from locally convex to locally concave side. Thus, if H h is the boundary of a convex body K of which x is an interior point, we must have j h (x) = 1, whereas i h (x) = 1 or i h (x) = 1 depending on whether u points inward or outward from K at x h (u) 2 H h = @K, (u 2 S 2 ). Now, the j h -index corresponds to the transverse orientation of H h that is such that whenever x h (u) is a simple regular point of H h , then the normal line to H h at x h (u), is oriented in the direction that j h decreases by one unit. Contrary to the usual transverse orientation of H h , it is clear from its de…nition that this transverse orientation of H h does not depend on the choice of the orientation of normal lines to x h (S 2 ) = x e h (S 2 ) : We call it 'the absolute transverse orientation'of H h . From the above, this absolute transverse orientation cannot change on an elliptic region (i.e., a region on which the Gauss curvature of H h remains positive): the absolute transverse orientation is then simply given by the direction of convexity. For our present study, the crucial point will be that a hedgehog of R 3 may admit reversals of its absolute transverse orientation along certain of its self-intersection curves formed by double hyperbolic points. Such a reversal is possible along a self-intersection curve formed by double hyperbolic point, but not necessary: it depends on the global geometry of the hedgehog [START_REF] Martinez-Maure | New notion of index for hedgehogs of R 3 and applications[END_REF].

Absolute body of H h in R 3

We call absolute body of H h R 3 , and we denote by K h , the set

K h := H h [ Int (H h ) , where Int (H h ) := fx 2 R 3 n H h jj h (x) 6 = 0g.
Comparison to the body of F rh in R 3 Proposition 4. Let H h be a hedgehog of R 3 such that h 2 C 1 (S 2 ; R).

We have i F

rh j h on R 3 n (H h [ F rh ). Proof. Let x 2 R 3 n (H h [ F rh ). Let A be a connected component of S 2 n (h x ) 1 (f0g)
, and denote by b A the number of connected components of its boundary in S 2 . Now, denote by C A (x) (resp. S A (x)) the number of local extrema (resp. saddle points) of h x in A. We know that:

C A (x) S A (x) = 2 b A , since the Euler characteristic of A is given by (A) = 2 b A . Thus the number N A (x) = C A (x)+S A (x) of critical points of h x in A is such that N A (x) = b A + 2 (C A (x) 1) b A . Therefore N rh (x) = P A N A (x) P A b A = 2c h (x)
, where the sums are taken over the set of connected components of S 2 n (h x ) 1 (f0g). Thus:

i F rh (x) = 1 N rh (x) 2 1 c h (x) = j h (x) :
Corollary. If the mean of h 2 C 1 (S 2 ; R) on S 2 is equal to 0, then the absolute body K h of H h is included in the body K (F rh ) of its focal (since in this case (h x ) 1 (f0g) is nonempty for all x 2 R 3 ).

Proofs of Theorem 1 and Corollary 2

Proof of Theorem 1. We have assumed that N rh (u) does not meet the singular locus of F rh , and we want to prove that there exists some r 2 R such that x h r (u) is an at least double hyperbolic point of H h r . We thus consider the family (H h r ) r2R of the hedgehogs that are parallel to H h . Note that the point x h r (u) = rh (u) + (h (u) r) u describes the entire normal line N rh (u) when r describes the entire real line. Our proof relies on the comparative evolution of the absolute and relative transverse orientations of H h r at x h r (u) when r describes the entire real line. De…ne a function " u h : R ! f 1; 0; 1g as follows: put " u h (r) = 1 if both transverse orientations of H h r are well-de…ned and identical at x h r (u); put " u h (r) = 1 if both transverse orientations of H h+r are well-de…ned and opposite at x h r (u); …nally put " u h (r) = 0 if one the two transverse orientations cannot be de…ned at x h r (u). If x h r (u) is a simple elliptic point of H h r , the relative transverse orientation of H h r at x h r (u) is given by u, whereas the absolute one is given by the direction of convexity at x h r (u), that is by

R 1 h r + R 2 h r (u) u, with R 1 h r (u), R 2 h r (u)
being the principal radii of curvature of H h r at x h r (u). In other words, we have in this case:

" u h (r) = sgn R 1 h r + R 2 h r (u) = sgn R 1 h + R 2 h (u) 2r :
In fact this de…nition makes sense for any r 2 R such that R h r (u) > 0 (i.e., such that x h r (u) is an elliptic point of H h r ). Note we have thus

" u h (r) = ( 1 if r < R 1 h (u) 1 if r > R 2 h (u) (since R h r (u) = R 1 h r R 2 h r (u) = (R 1 h (u) r) (R 2 h (u) r) > 0 if r = 2 [R 1 h (u) ; R 2 h (u)]), so that " u h : R ! f 1; 0; 1g has a sign change on [R 1 h (u) ; R 2 h (u)]. If r 2 fR 1 h (u) ; R 2 h (u)g, then R h r (
u) = 0 so that x h r is a singular point of H h r , and thus a point of F rh (namely, c 1 rh (u) or c 2 rh (u)). Since N rh (u) does not contain any singular point of F rh , x h r (u) is then a regular point of a cuspidal edge x h r ( ) (see Subsect. 2.4), which is separating a hyperbolic region of H h r from an elliptic one, and a sign change of " u h cannot occur at such a point. In fact, we still could de…ne " u h (r) by " u h (r) = sgn R 1 h r + R 2 h r (u) in order to be consistent with the changes of the i h r and j h r indices at the points of the adjacent elliptic region.

Finally, let us come to the case where

r 2 ]R 1 h (u) ; R 2 h (u)[. We then have R 1 h r (u) = R 1 h (u) r < 0 and R 2 h r (u) = R 2 h (u) r > 0 so that x h r (u) is a hyperbolic point of H h r : R h r (u) = R 1 h r (u) R 2 h r (u) < 0. If x h r (u) is moreover a single point of H h r , then for any that is close enough to r in ]R 1 h (u) ; R 2 h (u)[, x h (u)
is a hyperbolic point of H h , and the con…guration of h x h (u) (R ), h x h (u) (f0g) and h x h (u) (R + ) on S 2 is qualitatively the same that the one of h x h r (u) (R ), h x h r (u) (f0g) and h x h r (u) (R + ), so that there is no sign change of " u h at r in such a case (see the relationships between i h (x), j h (x) and r h (x), r + h (x), c h (x) that we gave in Section 3 for all

x 2 R 3 n H h ). Therefore, if a sign change of " u h occurs at r 2 ]R 1 h (u) ; R 2 h (u)[, then the hyperbolic point x h r (u) of H h r is a multiple point of H h r . But, a sign change of " u h at r 2 ]R 1 h (u) ; R 2 h (u)
[ cannot be due to the fact that x = x h r (u) is also an elliptic point x h r (v) or a singular point x h r (v) of H h r , which must be a regular point of a cuspidal edge of H h r since by assumption N rh (u) does not meet Sing (F rh ). Indeed, if x = x h r (u) is such a point x h r (v), then there exists a a neighborhood V of v in S 2 such that the absolute transverse orientation of x h r (V) is fully determined by the direction of convexity on its elliptic part, so that no change of this absolute transverse orientation can be due to the crossing with the image under x h r of a neighborhood of u in S 2 . Thus, in the present case, if a sign change of " u h occurs at r then x h r (u) is an (at least double) hyperbolic point of H h r , which achieves the proof.

Our proof of Corollary 2 will make use of the following remark.

Remark. Let H h be a hedgehog such that h 2 C 1 (S 2 ; R). For any u 2 S 2 such that x = x h (u) 2 R 3 nF rh , denote by F (rh x ) the foliation of S 2 by the level lines of h x : S 2 ! R. For any such u 2 S 2 , we have:

(u is a saddle point of F(rh x )) () (x = x h (u) is a hyperbolic point of H h ):
Proof of corollary 2. Assume …rst that there exists some u 2 S 2 such that N rh (u) \ Sing (F rh ) = ?. We then know from Theorem 1 that there exists some r 2 R such that x h r (u) = x h (u) ru 2 N rh (u) is an (at least) double hyperbolic point of H h r : that is, there exists v 2 S 2 n fug, such that:

x h r (u) = x h r (v) , R h r (u) < 0, and R h r (v) < 0. If this point x = x h r (u) belongs to R 3 n F rh , then i F rh (x) = S (x) 2,
where S (x) is the number of saddle points of F (rh x ) (see (3) and the above remark), and thus

N rh (x) = 2 (1 i F rh (x)) 6.
In that case, knowing that N rh remains constant on each component of R 3 n F rh , there thus exists an open set formed by points of R 3 n F rh lying on at least 6 normal lines to H h . To reach the same conclusion in the case where x = x h r (u) 2 F rh , it su¢ ces to prove that, arbitrarily close to x, there exists points y 2 R 3 n F rh such that i F rh (y) = S (y) 2, and thus N rh (y) = 2 (1 i F rh (y)) 6. Since u, v 2 S 2 are such that u 6 = v, R h r (u) < 0 and R h r (v) < 0, there exist U I, V J neighborhoods of respectively (u; r) and (v; r) in S 2 R, such that U \ V 6 = ? and R h (!) < 0 for all (!; ) in (U I) or (V J ). Besides, there exists a neighborhood

Y of x = x h r (u) = x h r (v) in R 3 such that any point y 2 Y\ (R 3 n F rh ) can be written under the form x h (!) with (!; ) 2 U I (resp. (!; ) 2 V J ). Therefore, any y 2 Y\ (R 3 n F rh ) is such that i F rh (y) = S (y)
2, and thus N rh (y) = 2 (1 i F rh (y)) 6. Indeed, the foliation of S 2 by the level lines of h y : S 2 ! R has at least two distinct saddle points: one in U and the other in V. Thus, in that case also, there exists an open set formed by points of R 3 n H h lying on at least 6 normal lines to H h .

Therefore, it only remains to consider the case where all the normal lines to H h meet the singular locus Sing (F rh ) of its focal F rh , that is: 8u 2 S 2 , N rh (u) \ Sing (F rh ) 6 = ?. But in this last case, there must exist a point of R 3 lying on in…nitely many normal lines to H h .

Critical saddle points

It is worth to stress that x = x h r (u) = x h r (v), the double hyperbolic point of H h r that Theorem 1 states to exist, is not an arbitrary double hyperbolic point of H h r but a double hyperbolic point of H h r in the vicinity of which the absolute transverse orientation of H h r is reversed. This property can be read on the foliation F (rh x ) of S 2 by the level lines of h x : S 2 ! R. Saying that x = x h r (u) = x h r (v) is a double hyperbolic point of H h r is equivalent to saying that u and v are saddle points of F (h x ) lying on (h x ) 1 (frg). Now, saying that x is a double hyperbolic point of H h r in the vicinity of which the absolute transverse orientation of H h r is reversed is equivalent to saying that these saddle points u, v of F (rh x ) lying on (h r) 1

x f0g = (h x ) 1 (frg) are 'critical saddle points'in the following sense.

De…nition. Let H h be a hedgehog of R 3 such that h 2 C 1 (S 2 ; R), and let u 2 S 2 be such that x = x h (u) is a hyperbolic point of H h . Note that u 2 S 2 is such that any v 2 S 2 that is su¢ ciently close to u is a saddle point of F (rh y ), where y = x h (v). We say that such a point v disconnects (h y ) 1 (R ) (resp. (h y ) 1 (R + ) if (h y ) 1 (R ) and (h y ) 1 (R ) n fvg resp. (h y ) 1 (R + ) and (h y ) 1 (R + )nfvg does not have the same number of connected components. Now, the saddle point u of F (rh x ) is said to be critical if the disconnecting or nondisconnecting character of v with respect to (h y ) 1 (R ) and (h y ) 1 (R + ) changes in every neighborhood of u in the region of (R h ) 1 R containing it.

By considering carefully the following three types of pairs of saddle points of F (rh x ) shown in Figure 4, we check that u and v are critical saddle points of F (rh x ) in the …rst case (a), and only in this case.

(a) (b) (c)

Recall that the critical and singular sets, C rh and S rh , can respectively be de…ned by

C rh := f(x; u) 2 U R 3 jr (h x ) (u) = 0g = f(x; u) 2 U R 3 jx 2 N rh (u)g ,
where N rh (u) := frh (u)g + Ru is the normal line to H h at x h (u), and

S rh := (x; u) 2 C rh 9k 2 [ j1; 2j ] ; x = c k rh (u) = f(x; u) 2 C rh j(det r 2 ) (h x ) (u) = 0g .
The following subset of C rh n S rh played an important role in our proof of Theorem 1:

c rh =: f(x; u) 2 C rh ju is a critical saddle point of F (rh x )g .
More precisely, our proof relies on the fact that under the assumption N rh (u) \ Sing (F rh ) = ?, the focal segment [c 1 rh (u) ; c 2 rh (u)] contains some interior point x such that (x; u) 2 c rh . Indeed, for every x 2 N rh (u), (x; u) 2 c rh if, and only if, there exist some r 2 R such that x = x h r (u) is a hyperbolic point of H h r and a neighborhood U of u on S 2 such that the absolute transverse orientation of x h r (U) is reversed in the vicinity of x h r (u). From this remark we can deduce the following re…nement of Corollary 2.

Proposition 5. Let H h be a hedgehog of R 3 such that h 2 C 1 (S 2 ; R). If there does not exist a point of R 3 lying on in…nitely many normal lines to H h , then there exist in…nitely many normal lines to H h that do no meet the singular locus of F rh and any one of these lines meets the closure of Int (F 1 rh ) \ Int (F 2 rh ).

Proof Let u be any point of S 2 such that N rh (u)\Sing (F rh ) = ?. For such a point u, we know from our proof of Theorem 1 that there exists some r 2 R such that u is a critical saddle point of F (rh x ), where x = x h r (u). By the very de…nition of a critical saddle point, this shows that C m (x) 2 and C M (x) 2. If x 2 R 3 nF rh , this exactly means that x 2 Int (F 1 rh ) \ Int (F 2 rh ). Now if x 2 F rh , we know that arbitrarily close to x, there are points y 2 R 3 nF rh , and that if y 2 R 3 nF rh is closed enough to x, then C m (y)

2 and C M (y) 2 (the arguments are the same as those used by E. Heil in [START_REF] Heil | Existenz eines 6-Normalenpunktes in einem konvexen Körper[END_REF]), so that y 2 Int (F 1 rh ) \ Int (F 2 rh ). Therefore, in any case, x 2 Int (F 1 rh ) \ Int (F 2 rh ).

In higher dimension, our idea is to adopt the same type of approach.

The four-dimensional case

In order to follow a similar approach in the four dimensional case, we must overcome a series of di¢ culties. First of all, when considering a hedgehog H h in R 4 , we have to deal with two types of 'hyperbolic points', that is of points x = x h (u) in which the principal radii of curvature are non zero and not all of the same sign: a hyperbolic point x = x h (u) is of type 1 if R h (u) < 0, and then it corresponds to a type 1 saddle point u of h x (i.e. to a critical point u of index 1 of h x ); and a hyperbolic point x = x h (u) is of type 2 if R h (u) > 0, and then it corresponds to a type 2 saddle point u of h x (i.e. to a critical point u of index 2 of h x ). We will often say simply 'i-saddle of F (rh x )'instead of 'type i saddle point u of h x ', (i 2 [ j1; 2j ]). Besides, in the four dimensional case the usual transverse orientation of x h (S 3 ) = xe h (S 3 ) is no longer relative but absolute due to the even parity of the dimension of R 4 : it does not depend on the choice between h and e h as the support function (i.e. it does not depends on the choice of the orientation of the normal lines to the hypersurface). On the other hand, if the index r h de…ned by

r h (x) = r + h (x) r h (x) for all x 2 R n+1 n H h , where r h (x) (resp. r + h (x)
) is the number of connected components of S n n h 1 x (f0g) on which h x : S n ! R is negative (resp. positive), is none other that the usual index i h for n+1 = 3 (see above and [START_REF] Martinez-Maure | New notion of index for hedgehogs of R 3 and applications[END_REF]), it provides us with a new index and a new transverse orientation which is relative (to the choice of the orientation of the normal lines to x h (S 3 ) = xe h (S 3 )) for n + 1 = 4. We will call it 'the relative transverse orientation'.

New index and transverse orientation

Let H h be a hedgehog of R 4 such that h 2 C 1 (S 3 ; R). Above, we de…ned the index r h : R 4 n H h ! N as follows:

r h (x) = r + h (x) r h (x) for all x 2 R 4 n H h , where r h (x) (resp. r + h (x)
) denotes the number of connected components of S 3 n h 1

x (f0g) on which h x : S 3 ! R is negative (resp. positive). Of course, the index r h remains constant on each connected component of R 4 n H h . In particular, r h is equal to 0 on the unbounded component of R 4 n H h . It worth noting that the value of r h (x) must obviously decrease by one unit as x orthogonally crosses H h at a simple elliptic point x 0 = x h (u 0 ) in the direction of u 0 2 S 3 . When the crossing occurs at a simple hyperbolic point x 0 = x h (u 0 ), it is necessary to distinguish two cases depending on whether the saddle point u 0 of F (rh x 0 ) disconnects the connected component, say L u 0 , of (h x 0 ) 1 (f0g) containing it or not. We can imagine an example of each of these two con…gurations by rotating the two …gures shown in Figure 5 around a vertical axis passing through the saddle point u 0 in R 3 S 3 = R 3 [f1g. If L u 0 nfu 0 g is disconnected, as in the …rst example, we can check that the value of r h (x) decreases by one unit as x orthogonally crosses H h at x 0 = x h (u 0 ) in the direction of u 0 . We say then that x 0 is an active hyperbolic point of H h , and that u 0 is an active saddle point of F (rh x 0 ). If L u 0 n fu 0 g is connected, as in the second example where L u 0 is a torus pinched at u 0 , we can check that the value of r h (x) does not change as x orthogonally crosses H h at x 0 = x h (u 0 ). We say then that x 0 is a neutral hyperbolic point of H h , and that u 0 is a neutral saddle point of F (rh x 0 ). In short, the 'relative transverse orientation'of H h (that is, the one that is associated with the r h -index) is such that whenever x h (u) is a simple regular point of H h , then the normal line to H h at x h (u) is oriented in the direction of u (resp. u) if x h (u) is elliptical (resp. hyperbolic and active), and is not oriented if x h (u) is a neutral hyperbolic point.

Pairing saddle

Neutral saddle Of course, the same crossing but in the opposite direction results in the elimination of such a pair of critical points. Mathematically, such a transition can be regarded as a path in the vector space C 1 (S 3 ; R) that crosses a 'codimension 1 non-Morse stratum' transversely at one point. We know from 'parametrized Morse theory'that: (i) at the very moment of the transition, h x : S 3 ! R admits a so-called 'birth-death singularity'; (ii) the transition itself can locally be described by a smooth family of functions f t : u 7 ! f t (u) parametrized by t, (t 2 R), of the form

f t (u 1 ; u 2 ; u 3 ) = u 3 1 tu 1 + " 2 u 2 2 + " 3 u 2 3 + g (t)
, where " 2 ; " 3 2 f 1g, with respect to some local coordinates (u 1 ; u 2 ; u 3 ); when t < 0, f t : u 7 ! f t (u) is a Morse function without critical point; f 0 : u 7 ! f 0 (u) is a generalized Morse function with a birth-death singularity at u = (0; 0; 0); when t > 0, f t : u 7 ! f t (u) is a Morse function with two critical points of consecutive indices (these points were 'born'at t = 0).

When x 2 R 4 transversally crosses F rh at a simple regular point in the direction of the transverse orientation, we are thus in one of the following three cases:

Case " 2 = " 3 = 1 " 2 6 = " 3 " 2 = " 3 = 1 Sheet crossed by x F 1 rh F 2 rh F 3 rh
Pair of critical points that are born 1 local minimal and 1 type 1 saddle 1 type 1 saddle and 1 type 2 saddle 1 local maxima and 1 type 2 saddle

In the …rst (resp. third) case, a 'trivial bubble' (a 'trivial centresaddle pairing' of F (rh x )) was born as shown in Figure 6 (a): the centre of this trivial centre-saddle pairing is of course located inside the bubble, and the corresponding saddle point is obviously active. In the second case, two saddle points of distinct types of F (rh x ) were born. In the vicinity of each of them, the level set of the corresponding saddle point looks qualitatively like the piece of surface shown in Figure 6 (b). Moreover, the two level sets corresponding to these two saddle points are in fact two pinched torus that are 'linked somehow like a Hopf link' [19]. In particular, these two saddle points are neutral.

Let x 2 R 4 n F rh . For any regular value d of h x : S 3 ! R, denote by g hx (d) the genus of the (not necessarily connected) surface (h x ) 1 (fdg). Now, for any critical value c of h x , put g hx (c) := min ">0 (max (fg hx (d) jd regular value of h x in ]c "; d + "[ g)) , and then

G h (x) = 1 2 X c2C h (x)
g hx (c) , where C h (x) = h x u 2 S 3 jrh x (u) = 0 .

From the above study, when x transversally crosses F rh at a simple regular point x 0 2 F rh , the value of G h (x) changes if, and only if x 0 2 F 2 rh . In this case, the value of G h (x) increases (resp. decreases) by one unit if the crossing occurs in the direction of the transverse orientation of F rh (resp. in the opposite direction). Knowing that G h (x) = 0 when x is far from H h , we deduce the following result. Theorem 3. Let H h be a hedgehog of R 4 such that h 2 C 1 (S 3 ; R), let F rh denote its focal surface, and let F 2 rh be the second sheet of F rh . We have:

8x 2 R 4 n F 2 rh , i F 2 rh (x) = G h (x).
As a corollary we obtain that:

8x 2 R 4 n F 2 rh , Int F 2 rh = x 2 R 4 n F

Critical saddle points

As in dimension 3, our idea will be to focus our attention, for each u 2 S 3 such that N rh (u) \ Sing (F rh ) = ?, on the values of r such that x = x h r (u) is a (type 1 or type 2) hyperbolic point of H h r (and thus, u a saddle point of F (rh x )) at which the relative transverse orientation of H h r switches (here, between the active and neutral modes): we will say then that u is a critical (type 1 or type 2) saddle point of F (rh x ), where x = x h r (u).

De…nition. Let H h be a hedgehog of R 4 such that h 2 C 1 (S 3 ; R), and let x = x h (u) be a hyperbolic point of type i of H h , (i 2 [ j1; 2j ]). Then, any v 2 S 3 that is su¢ ciently close to u on S 3 is a i-saddle point of F (rh y ), where y = x h (v). We say that such an v disconnects (h y ) 1 (f0g) if (h y ) 1 (f0g) and (h y ) 1 (f0g) n fvg do not have the same number of connected components. Now, the saddle point u of F (rh x ) is said to be critical if the disconnecting or nondisconnecting character of v changes in every neighborhood of u in the region of (R h ) 1 (R ) containing it.

Return brie ‡y to singularity and Morse theories viewpoints. The critical and singular sets, C rh and S rh , can respectively be de…ned by

C rh := f(x; u) 2 U R 4 jr (h x ) (u) = 0 g = f(x; u) 2 U R 4 jx 2 N rh (u)g ,
where N rh (u) := frh (u)g + Ru is the normal line to H h at x h (u), and

S rh := (x; u) 2 C rh 9k 2 [ j1; 3j ] ; x = c k rh (u) = f(x; u) 2 C rh j(det r 2 ) (h x ) (u) = 0g .
For each i 2 [ j1; 2j ], consider the following subsets of C rh n S rh :

i (rh) := f(x; u) 2 C rh n S rh ju is a i-saddle point of F(rh x )g , and c i (rh) := f(x; u) 2 i (rh) j u is a critical i-saddle point of F(rh x )g . For each i 2 [ j1; 2j ], i (rh) n c i (rh) is the following union of disjoint open subsets: i (rh) n c i (rh) = a i (rh) F n i (rh)
, where: a i (rh) := f(x; u) 2 i (rh) j u is an active i-saddle point of F(rh x )g and n i (rh) := f(x; u) 2 i (rh) j u is a neutral i-saddle point of F(rh x )g . More precisely, for each i 2 [ j1; 2j ], c i (rh) separates a i (rh) and n i (rh) in i (rh). We are now prepared to prove our Theorem 4.

Proofs of Theorem 4 and Corollary 3

Proof of Theorem 4. Since N rh (u) \ Sing (F rh ) = ?, the 3 centers of principal curvature c 1 rh (u), c 2 rh (u) and c 3 rh (u) are pairwise distinct.

When r 2 ]R 1 h (u) ; R 2 h (u)[ tends to R 1 h (u), u is an active 1-saddle point of F(rh x ), where x = x h r (u), whereas when r 2 ]R 1 h (u) ; R 2 h (u)[ tends to R 2 h (u), u is a neutral 1-saddle point of F(rh x ), where x = x h r (u). More precisely, (x h r (u) ; u) 2 a i (rh) (resp. (x h r (u) ; u) 2 n i (rh)) when r 2 ]R 1 h (u) ; R 2 h (u)[ tends to R 1 h (u) (resp. R 2 h (u)).
From this it follows that there exists some r

1 2 ]R 1 h (u) ; R 2 h (u)[ such that u is a critical 1-saddle point of F(rh x 1 )
, where x 1 = x h r 1 (u). Since u is a critical 1-saddle point of F(rh x 1 ), there exists some v 2 S 3 n fug that is a critical point of F(rh x 1 ) lying on the connected component, say L u , of the level set of h x 1 that contains u. Such an v is of course such that

x 1 = x h r 1 (v). Since N rh (u) \ Sing (F rh ) = ?, if v is a degenerate critical point of (h r 1 ) x 1 , then v is such that R h r 1 (v) = 0 (since v is degenerate), rR h r 1 (v) 6 = 0 and rk [T v x h r 1 ] = 2 (otherwise x 1 would belong to Sing (F rh )).
It follows that under our assumptions, H h r 1 has then a cusp singularity at x 1 = x h r 1 (v); that is, the image under x h r 1 of a neighborhood V of v is di¤eomorphic to a neighborhood of 0 in f(x 1 ; x 2 ; x 3 ; x 4 ) 2 R 4 jx 2 2 = x 3 1 g. But in such a con…guration, the relative transverse orientation of x h r 1 (v) is fully determined in the vicinity of x 1 by the index of the principal radius of curvature that vanishes at v: indeed, if R 2 h r 1 (v) = 0, then the hyperbolic points of x h r 1 (V) are neutral in the vicinity of x 1 , and if R 1 h r 1 (v) = 0 (resp. R 3 h r 1 (v) = 0 the relative transverse orientation of x h r 1 (V) at a regular point x h r 1 (w) that is close enough to x 1 = x h r 1 (v) is given by:

sgn [R h r 1 (w)] w (resp. sgn [R h r 1 (w)] w) .
Thus if v is a degenerate critical point of (h r 1 ) x 1 , then no switch of the relative transverse orientation (between the active and neutral modes) at u can be due to the crossing of x h r 1 (V) with the image under x h r 1 of a neighborhood of u in S 3 . Therefore, v is a nondegenerate critical point of (h r 1 ) x 1 and thus a 1-saddle point of F(rh x 1 ) by the way it is obtained. Thus r

1 2 ]R 1 h (u) ; R 2 h (u)[ is such that x h r 1 (u) = x h (u) r 1 u is an (at least) type 1 double hyperbolic point of H h r 1 .4
Finally, we can adapt the above arguments to prove that there exists some r

2 2 ]R 2 h (u) ; R 3 h (u)[ such that x h r 2 (u) = x h (u) r 2 u is an (at least) type 2 double hyperbolic point of H h r 2 .
Proof of Corollary 3. Let u 2 S 3 such that N rh (u)\Sing (F rh ) = ?. For such a point u, we know from our proof of Theorem 4 that there exists (r 1 ; r 2 ) 2 R 2 such that, for each i 2 [ j1; 2j ], x h r i (u) = x h (u) r i u is an (at least) type i double hyperbolic point of H h r i , and u a critical saddle point of F (rh x i ), where x i = x h r i (u).

Assume …rst that x 1 = x h (u) r 1 u is in R 4 n F rh . Since x 1 is an (at least) double type 1 hyperbolic point, we then have S 1 (x 1 )

2. Now (see Subsubsection 2.3.2)

2 P k=1 i F k rh (x 1 ) = (1 C m (x 1 )) + (C m (x 1 ) S 1 (x 1 ) 1) = S 1 (x 1 ) ,
and consequently

i F rh (x 1 ) = 3 P k=1 i F k rh (x 1 ) = S 1 (x 1 ) + i F 3 rh (x 1 ) S 1 (x 1 ) . Therefore i F rh (x 1 )
2, and thus N rh (x 1 ) = 2 (1 i F rh (x 1 )) 6. Now, if x 1 2 F rh , we can proceed as in the proof of Corollary 1 to show that arbitrarily close to x 1 , there exists some y 1 2 R 4 n F rh such that S 1 (y 1 ) 2, and thus N rh (y 1 ) = 2 (1 i F rh (y 1 )) 6. Therefore, in any case, x 1 = x h (u) r 1 u is in the closure of fx 2 R 4 n F rh jN rh (x) 6g.

But, in fact, we can be more precise. Since u a critical saddle point of F (rh x 1 ), we know that arbitrarily close to x 1 , there exists some y 1 2 R 4 n F rh that is a neutral hyperbolic point and thus a point of Int (F 2 rh ). Furthermore, since u is a critical saddle point of F (rh x 1 ), we have C m (x 1 ) 2, and we know [START_REF] Heil | Existenz eines 6-Normalenpunktes in einem konvexen Körper[END_REF] that arbitrarily close to x 1 , there are points y 1 2 R 3 n F rh , and that if y 1 2 R 3 n F rh is close enough to x, then C m (y 1 ) 2 and thus y 1 2 Int (F 1 rh ). Therefore, arbitrarily close to x 1 , there exists some y 1 2 R 4 nF rh such that y 1 2 Int (F 1 rh )\Int (F 2 rh ). Of course, we can prove in the same way that x 2 = x h (u) r 2 u is also in the closure of fx 2 R 4 n F rh jN rh (x) 6 g, and more precisely in the closure of Int (F 2 rh ) \ Int (F 3 rh ).

5 Further results and remarks

Minimal spherical shell of a convex body

As we said in introduction, E. Heil proved the concurrent normal conjecture in R 2 (resp. R 3 ) using of the existence of a minimal spherical shell for any convex body of R 2 (resp. R 3 ) which had been established by T. Bonnesen [START_REF] Bonnesen | Über das isoperimetrische De…zit ebener Figuren[END_REF] (resp. N. Kritikos [START_REF] Kritikos | Über konvexe Flachen und einschließ ende Kugeln[END_REF]). In 1988, I. Bárány extended this existence to higher dimensions [START_REF] Bárány | On the minimal ring containing the boundary of a convex body[END_REF]. Given a convex body K in R n+1 , he de…ned r (x) := max (fr 2 R + jB (x; r) K g) ;

and R (x) := min (fr 2 R + jK B (x; r)g) , where B (x; r) denotes the closed ball with center x and radius r, and proved the following:

Theorem (I. [START_REF] Bárány | On the minimal ring containing the boundary of a convex body[END_REF]. There exists a unique point x 0 in K at which the function R r : K ! R + , x 7 ! R (x) r (x) attains its minimum value.

The set C (x 0 ; r (x 0 ) ; R (x 0 )) := fx 2 R n+1 jr (x 0 ) kx x 0 k R (x 0 )g is called the minimal spherical shell of K. On each sphere bounding C (x 0 ; r (x 0 ) ; R (x 0 )) there are at least two points of K. More precisely, C (x 0 ; r (x 0 ) ; R (x 0 )) can be characterized as follows.

Theorem (I. [START_REF] Bárány | On the minimal ring containing the boundary of a convex body[END_REF]. The point x 0 2 K is the center of the minimal spherical shell of K if, and only if, there exist (u 1 ; : : : ; u p ) (S n ) p and (v 1 ; : : : ; v q ) 2 (S n ) q , (p; q 1), such that:

(i) Conv (fu 1 ; : : : ; u p g) \ Conv (fv 1 ; : : : ; v q g) 6 = ?, where Conv denotes the convex hull;

(ii) x 0 +r (x 0 ) u i 2 @K and x 0 +R (x 0 ) v j 2 @K for all (i; j) 2 [ j1; pj ] [ j1; qj ] ;

where @K denotes the boundary of K.

Extension to hedgehogs

Any hedgehog H h of R n+1 that is such that h 2 C 2 (S n ; R) can be regarded as a parallel hypersurface to the boundary of some convex body of class C 2 + , say K, in R n+1 (see 2:1:1): there exists 2 R such that h = h K , where h K is the support function of K. We can thus de…ne the minimal spherical shell of H h to be the transversely oriented shell that is bounded by the transversely oriented spheres or point spheres S (x 0 ; r (x 0 )

) and S (x 0 ; R (x 0 ) ) ,

where C (x 0 ; r (x 0 ) ; R (x 0 )) is the minimal spherical shell of K, and where S (a; r)) denotes the sphere of radius jrj centered at a that is transversely oriented by its outward (resp. inward) pointing normals if r > 0 (resp. r < 0) holds, and the point sphere fag if r = 0 holds. This de…nition is of course independent from the choice of the convex body K of class C 2 + the boundary of which is parallel to H h .

Remark. All these minimal spherical shells can be interpreted in terms of pedal hypersurfaces. Recall that the pedal hypersurface of the hedgehog H h R n+1 with respect to a point x 2 R n+1 is parametrized by

S n ! R n+1 u 7 ! x + h x (u) u,
where h x (u) = h (u) hx; ui. In other words, to any u 2 S n is associated the foot of the perpendicular line from x to the support hyperplane H h at x h (u). If C (x 0 ; r (x 0 ) ; R (x 0 )) is the minimal spherical shell of some convex body K with support function h 2 C 2 (S n ; R) in R n+1 , we can easily check that: (i) the pedal hypersurface of the hedgehog H h (r(x 0 )+R(x 0 ))=2 with respect to x 0 is contained in the closed ball B x 0 ; R(x 0 ) r(x 0 ) 2 ; (ii) no other pedal hypersurface of H h is contained in a smaller closed ball of R n+1 ; (iii) Properties (i) and (ii) permit to characterize the minimal spherical shell of K. Moreover, this interpretation extends to hedgehogs. Let us give an example. Figure 7 shows: (a) the ellipse with support function 2 S 1 = R=2 Z 7 ! h ( ) := p 9 cos 2 + 25 sin 2 ; the minimal spherical shell of which is C (0 R 2 ; 3; 5) where 0 R 2 is the origin of R 2 ; (b) the hedgehog with support function h (r (0 R 2 ) + R (0 R 2 )) =2 = h 4, which is parallel to the ellipse ; (c) the pedal P 0 (H h 4 ) of H h 4 with respect to the origin; (d) all these curves together with the circle C (0 R 2 ; 1) with center 0 R 2 and radius 1 2 (5 3) = 1. Note that in this example, the minimal spherical shell C (0 R 2 ; 1; 1) of H h 4 is formed by two circles that are the same but with opposite transverse orientations. Figure 8 shows the hedgehog with support function h , and these curves with the m.s.s. of H h 7 2

Bárány's characterization theorem of the minimal spherical shell can of course be adapted to hedgehogs: Theorem 5. Let H h be a hedgehog of R n+1 such that h 2 C 2 (S n ; R), and let x 2 R n+1 and (r; R) 2 R 2 be such that r h x R, where h x (u) = h (u) hx; ui, (u 2 S n ). Then C (x; r; R) is the minimal spherical shell of H h if, and only if, there exist (u 1 ; : : : ; u p ) (S n ) p and (v 1 ; : : : ; v q ) 2 (S n ) q , (p; q 1), such that:

(i) Conv (fu 1 ; : : : ; u p g) \ Conv (fv 1 ; : : : ; v q g) 6 = ?, where Conv denotes the convex hull;

(ii) h x (u i ) = r, h x (v j ) = R, and rh x (u i ) = rh x (v j ) = 0 : for all (i; j) 2 [ j1; pj ] [ j1; qj ]. Now, we can construct h 2 C 2 (S 3 ; R) such that: (i) jhj 1; (ii) the set fu 2 S 3 jh (u) = 1g has exactly two distinct elements u 1 , u 2 , and the set fu 2 S 3 jh (u) = 1 g has exactly two distinct elements v 1 , v 2 ; (iii) [u 1 u 2 ] \ [v 1 v 2 ] 6 = ?; (iv) h has exactly six critical points: 2 minima, 2 maxima and two saddle points. Let (e 1 ; e 2 ; e 3 ; e 4 ) be the canonical basis of R 4 . Put u 1 = e 1 , u 2 = e 1 , v 1 = e 4 ; v 2 = e 4 , w 1 = e 2 , w 2 = e 2 , and identify S 3 nfe 4 g to R 3 through the stereographic projection from e 4 onto the hyperplane V ect (e 1 ; e 2 ; e 3 ). The point v 1 = e 4 is thus identi…ed with the origin of R 3 . We then see that we can choose h so that h (u 1 ) = h (u 2 ) = 1, h (w 1 ) = 1=2; h (w 2 ) = 1=2, and h (v 1 ) = h (v 2 ) = 1, such that the foliation of R 3 generated by the level surfaces of h is made by leaves di¤eomorphic to S 2 , with the exception of the leaves shown in Figure 9 (on which the dashed line is the circle S 2 \ V ect (e 1 ; e 2 )). On the …rst picture of Figure 9, the curve has to be rotated around the axis Re 2 .

Figure 9 For such a h 2 C 2 (S 3 ; R), C (0 R 4 ; 1; 1) is the minimal spherical shell of H h , and there are only six normal lines to H h passing through 0 R 4 . Therefore: Theorem 6. It is not true that for any convex body K of R 4 , there are at least 8 normal lines passing through the center of the minimal spherical shell of K.

Figure 2 .

 2 Figure 2. Plane hedgehog with C 2 -support function

Figure 3 .

 3 Figure 3. Generic singularities of hedgehogs in R 3

Figure 5 4. 2 2 rh

 522 Figure 5

2

 2 rh jG h (x) 6 = 0 ; and thus:x 2 Int F 2 rh ()h x : S 3 ! R admits a smooth level surface with nonzero genus ;which is the statement of theorem 2.(a) (b) Figure6

7 2 2 .Figure 7 .Figure 8 . H h 7 2 , P 0 H h 7 2

 727822 Figure 7. (a) The ellipse (b) H h 4 (c) P 0 (H h 4 ); (d)These curves and C (0 R 2 ; 1)

  in terms of the Betti numbers of S 2p+1 , can simply be rewritten as: 8k 2 [ F k rh (x) negatively measures how far from equality we are in the k th strong Morse inequality when considering the Morse function h

	j1; 2p + 1j ], i F k rh (x) These 2p + 1 indices are nonpositive, and, for every k 2 [ j1; 2p + 1j ], 0. the index i

Acknowledgment

The author is very grateful to the anonymous referee for careful reading of the manuscript and for valuable suggestions that signi…cantly helped to improve the manuscript.


Rashad Ghassani
email: rashad.ghassani@gipsa-lab.grenoble-inp.fr

Antoneta Iuliana Bratcu

Remus Teodorescu

Genetic Algorithm Applied to State-Feedback Control Design of Grid and Circulating Current in Modular Multilevel Converters

Keywords: modular multilevel converter, cascaded control, state-feedback, genetic algorithm

This paper discusses the application of a genetic algorithm (GA) to control system design for Modular Multilevel Converters (MMCs). In particular, genetic algorithm is used to compute the gains of a state-feedback controller for multi-input/multi-output (MIMO) plant model. This GA-optimized state-feedback controller is used to control both grid and circulating current of the MMC. This assures that the two currents' input-coupled dynamics are managed using a MIMO strategy. A detailed MATLAB ® /Simulink ® model of a three-phase MMC is further used to validate the proposed control technique. Different simulations show that the GA-optimized state-feedback controller outperforms the conventional cascaded control.

INTRODUCTION

With the fast improvement of wind farms, the request for high-power, high-quality transmission systems is becoming more urgent. Modular Multilevel Converter (MMC) which is based on high-voltage direct-current (HVDC) innovation gives a promising solution [START_REF] Lesnicar | An innovative modular multilevel converter topology suitable for a wide power range[END_REF]). In recent years, MMCs have become one of the most appealing topologies for high-voltage industrial applications such as HVDC, medium-voltage motor drives, and energy storage systems. The modularity, flexible expandability, transformer-less architecture, and low cost of MMCs have all contributed to their widespread acceptance in the industry.

Controlling the transferred power through the MMC to the grid is the main objective. Two control structures have been proposed in the literature, which are multivariable [START_REF] Munch | Integrated current control, energy control and energy balancing of modular multilevel converters[END_REF]) and cascaded [START_REF] Bergna Diaz | An energy-based controller for hvdc modular multilevel converter in decoupled double synchronous reference frame for voltage oscillation reduction[END_REF]) approaches. The first approach allows to handle the MMC state-space equations as a nonlinear multi-variable model. It requires some advanced control strategies in order to derive the associated control law and perform stability analysis. The second approach, which is based on different time scales assumptions, treats the system in a decoupled way, such that the control algorithm is created and modified step by step [START_REF] Yijing | Control induced explicit time-scale separation to attain dc voltage stability for a vsc-hvdc terminal[END_REF]).

The cascaded approach in dealing with all the different variables of MMC MIMO plant is the most preferred approach in the industry. The control design approach to MIMO systems requires the state space representation, in which the state feedback control is used. The most commonly used strategy for state-feedback control design is the Linear Quadratic Regulation (LQR) [START_REF] Solihin | Comparison of lqr and pso-based state feedback controller for tracking control of a flexible link manipulator[END_REF]). Despite the positive results produced by this approach, control design is a difficult process due to the trial-anderror method used in the weight matrix creation where hard tuning the controller parameters could be tedious task. However, using Artificial Intelligent (AI) control design techniques can be a promising alternative. Controldesign approaches make use of this domain knowledge for the control application to come up with a control system that minimizes some weighted performance function. The genetic algorithm (GA) is heuristic search method that holds a great potential in the control system design problem [START_REF] Sivanandam | Introduction to Genetic Algorithms[END_REF]).

This paper proposes a new optimal control strategy for a MIMO state-feedback controller that controls both grid and circulating current using GA to find the controller gains. The state-feedback optimization problem is solved under the balanced-grid model, but it is tested under the unbalanced conditions. This paper is organized as follows. Section 2 introduces the balanced-grid three-phase MMC model, which is used later to design the proposed GA-optimized MIMO state-feedback controller in Section 3. Section 4 discusses the GA setup and calibration. Section 5 shows the results obtained on a comprehensive MATLAB ® /Simulink ® three-phase MMC model. The conclusion is presented in the last Section 6.

MMC CURRENT STATE EQUATIONS

A detailed diagram of a typical three-phase MMC is shown in Figure 1. Each arm consists of N sub-modules (SM), the arm resistance (adding the losses within each arm) and arm inductance. The voltage of each SM, that is composed of a two semiconductors switches and a capacitor, is defined by v k,j , where subscript k represents the converters arms (k = u, l, upper arm and lower arm respectively) and subscript j represents phase (j = a, b, c). On the ac side, the converter is assumed to be connected to the grid. The dc side of converter is grounded at its midpoint. Using Kirchhoff voltage law (KVL), the dynamic equations of MMC system in each phase can be expressed by:

v d 2 -v u -Ri u -L di u dt = v a -v d 2 + v l + Ri l + L di l dt = v a (1)
In Figure 1, each sub-module capacitor is represented by an (A-B) Block. Different MMC parameters can be summarized as follows: dc variables (current and voltage) are represented by the subscript "d", ac variables for each phase are denoted by v a,b,c . The number of sub-modules in each arm is N , while the resistance and inductance of the arm are R and L, respectively. For each phase, the voltage for a single SM i is denoted as v i cu,l for the upper and lower arm with an insertion index defined as n i u,l = 1 if the SM is inserted and 0 otherwise. The total arm voltage can be defined as total number of inserted capacitor SMs such that v u,l = N i=1 n i u,l • v i cu,l and the sum of capacitor voltages on an arm is defined as v Σ cu,l = N i=1 v i cu,l , while the insertion index in an arm is defined as n u,l = (1/N ) N i=1 n i u,l . Assuming that all capacitor voltages are equal, and combining the above-mentioned formulas, the following conclusion holds: v u,l = n u,l • v Σ cu,l . Therefore, the output current i s and output voltage v s can be written as:

i s = i u -i l , v s = (v u -v l )/2
(2) and the circulating current i c and internal voltage v c can be defined as:

i c = (i u + i l )/2, v c = (v u + v l )/2
(3) Based again on Figure 1 andEquations (1), (2), and (3), the dynamic equations of MMC for phase j can be expressed by:

ic = -R/L • i c -1/(2L) • v u -1/(2L) • v l + 1/(2L) • v d i s = R/L • i s -1/L • v u -1/L • v l -2/L • v a (4)
Replacing v s from ( 2) and v c from (3), ( 4) is obtained.

It can be shown that the circulating current i c follows a first-order dc-dynamics and output current i s obeys a firstorder ac-circuit dynamics.

ic = -R/L • i c -v c /L + 1/(2L) • v d (5) i s = -R/L • i s + 2/L • v s -2/L • v a (6)
The two currents can be regulated individually using proportional integral (PI) or proportional-resonant (PR) linear controllers by employing v c in (2) and v s in (3) as control inputs [START_REF] Sharifabadi | Design, Control and Application of Modular Multilevel Converters for HVDC Transmission Systems[END_REF].

STATE-FEEDBACK CONTROL DESIGN

In this section, the state-feedback controller [START_REF] Bratcu | State-feedback control of grid and circulating current in modular multilevel converters[END_REF]), is introduced. Starting from the state equations ( 5) and ( 6) characterizing the dynamics of the two currents on a single phase, a new control design can be made. It is worth noting that these dynamics are intertwined at the input level: upper and lower-arm voltages intervene in both, which will now be the control inputs. On the second-order system, dc voltage v d acts as a constant disturbance, whereas voltage v a is perceived as a grid-frequency sinusoidal disturbance.

ic = -R/L • i c -1/(2L) • v u -1/(2L) • v l + 1/(2L) • v d is = R/L • i s -1/L • v u -1/L • v l -2/L • v a (7)
For the MIMO plant ( 7), a full-state feedback is developed to achieve the necessary closed-loop dynamics [START_REF] Bratcu | State-feedback control of grid and circulating current in modular multilevel converters[END_REF]). In order to place the desired closed-loop dynamics and to achieve the required control objectives: ensuring a constant-reference tracking for the dc-component of i c , a ω-sinusoidal-reference tracking for i s , as well as a 2ω-sinusoidal disturbance rejection on i c .

To this end, five extra integral states were added to achieve the desired objectives as follows:

             ẋi1 = -x i2 + i * s -i s ẋi2 = ω 2 • x i1 ẋi3 = i * c -i c ẋi4 = -x i5 + i * c -i c ẋi5 = 4ω 2 • x i4 (8)
States x i1 and x i2 correspond to a resonant integrator on the grid frequency ω, state x i3 is that of an ordinary integrator that tries to eliminate the dc steady state error and finally, states x i4 and x i5 refer to a resonant integrator on double the grid frequency, 2ω. This combination between state-feedback control and resonant integrators can be seen as alternative to the classical integrators.

The MMC model can be represented by MIMO extended plant with x e = [i c i s x i1 x i2 x i3 x i4 x i5 ] T as state variables, u e = [v u v l ] T as control input vector and u p = [v d v a ] T as disturbance input vector. Thus, the whole MIMO system can be described as ẋe = A e • x e + B e • u e + B p • u p , where the state matrix A e and input matrices B e and B p are as follows:

A e =          -R/L 0 0 0 0 0 0 0 -R/L 0 0 0 0 0 0 -1 0 -1 0 0 0 0 0 ω 2 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 4ω 2 0          B e = -1/(2L) -1/L 0 0 0 0 0 -1/(2L) -1/L 0 0 0 0 0 T B p = 1/(2L) 0 0 0 0 0 0 0 -2/L 0 0 0 0 0 T
After ensuring that the extended system (A e , B e ) is controllable, a full state-feedback controller of the form u e = -K e • x e can be computed to impose the desired closed-loop dynamics [START_REF] Bratcu | State-feedback control of grid and circulating current in modular multilevel converters[END_REF]. Two of the new poles correspond to the original second-order dynamics of both currents, i c and i s , the other five poles correspond to the integral states. Vector control input is

u e = [v * u v * l ] T . Internal voltage v * c = (v * u + v * l )/2 and grid voltage v * s = (v * l -v * u )
/2 are computed, then sent as references to the modulation process, in order to obtain the desired upper and lower arm indices n * u and n * l , respectively.

The biggest challenge in this proposed state-feedback controller is how to choose the imposed closed-loop poles, because this choice will determine the control gain vector, K e . In the next section, a genetic algorithm technique is proposed to find the controller gains, which are optimal in view of a suitably defined criterion.

PROPOSED GENETIC-ALGORITHM-BASED

COMPUTATION OF GAINS Genetic Algorithms (GA) are based on Darwin's ideas of natural selection and evolutionary processes. Only the fittest individuals survive in a selection process, leaving the poor performers behind. The cost function is commonly referred to as a fitness function, and the process of 'survival of the fittest' entails a maximization approach. A GA begins by random population of initial solutions, and then applies a series of operations to generate a new population. Following that, the worst individuals are eliminated from the population, while the best ones are included in the next generation [START_REF] Sivanandam | Introduction to Genetic Algorithms[END_REF]).

Fitness Function

As stated earlier, the GA begins with a purely random initial population; in this case the initial population is a pool of the closed-loop poles to be optimized. Using some cost function each candidate solution (set of poles) is evaluated in the initial population, the top elite candidates are then transferred directly to the new generation, while the other individuals undergo different genetic algorithm operators such as crossover, in which two individuals from the initial population (parents) are reproduced to produce two new individuals (children), and mutation which yields a new intermediate generation. The next generation is formed by applying these same rules, and the process is repeated until a convergence criterion is reached.

Using GA-based techniques, a way to find the statefeedback controller gains to meet the design specifications is proposed in this work. The suggested approach is used to determine the appropriate state-feedback parameters and eliminate the time-consuming and repetitive trial-anderror procedure. As a result, no weight matrices must be chosen as in the LQR technique. The output current and circulating current tracking error are the chosen design specifications. So, it is proposed that the fitness function to take into account the circulating current and output current error as follows:

J = 1 N s Ns i=1 k 1 • |i ci -i * c | + 1 N s Ns i=1 k 2 • |i si -i * s | (9)
where k 1 and k 2 are weights to be adjusted, i * c and i * s are the circulating current and output current references respectively, while N s is the number of samples in the finite-horizon model. As previously stated, the proposed MIMO extended MMC plant contains 7 states and 2 control inputs, the "optimal" feedback matrix K e to be found is represented by 14 variables. The optimal closed-loop poles are found first using the proposed GA and after that the state-feedback control gain K e is calculated. Algorithm 1 presents the different steps of the optimization procedure. The crossover probability P c , mentioned in line 6, is equal to 0.9, while the mutation probability P m is chosen to be equal to 0.3. The chosen population size is 120 and the number of generations is set to 50. Since it is equally important to control both currents, the weight parameters k 1 and k 2 are fixed to 1. The different steps of the proposed GA can be summarized as follows: first of all, the initial purely random and bounded population P 0 is created. After that, the evaluation of all the candidate poles is started. This evaluation process is done by running offline the complete MMC simulation model presented in Appendix A for a time of 1 s under normal grid conditions, by using the candidate poles to be evaluated. As it is run off-line, this algorithm can work for any number of submodules per arm, N . This process is then repeated for all the population.

RESULTS AND NUMERICAL SIMULATIONS

The genetic algorithm optimization effectively identified the MIMO optimum controller's feedback gains after passing through 50 generations and calculating for around two hours. The 7-variable vector of the imposed closed-loop poles results from the proposed genetic algorithm. The feedback matrix K e is then easily computed using pole placement, while K e 3phase , the three-phase extension of single phase gain K e , is used as the feedback gain matrix to the optimized state-feedback controller.

A detailed MATLAB ® /Simulink ® model of a three-phase MMC, whose parameters are presented in Appendix A, is used to validate the suggested control technique. The traditional control system, which is commonly used in practice and is based on many cascaded control levels [START_REF] Sharifabadi | Design, Control and Application of Modular Multilevel Converters for HVDC Transmission Systems[END_REF], is used as a baseline here. Indeed, the suggested state-feedback controller substitutes the two distinct control loops of i c and i s , while preserving other control levels present in the global multiple-level-based control method. As a result, the energy control level, as well as PWM modulation implementation, are preserved. The chosen scenario to test the proposed approach is set at 1.3 s, with an imbalance in grid conditions starting at time 0.7 s and ending at time 1.1 s. The balanced grid is defined by grid voltage positive sequence v g pos = 1 p.u. and a voltage negative sequence v g neg = 0 p.u., while the unbalance grid is characterized by a positive sequence v g pos = 0.8 p.u. and a negative sequence v g neg = 0.2 p.u..

GA-optimized state-feedback control results

The MIMO full-state feedback based on genetic algorithm is compared with the conventional cascaded control under the same relevant scenarios. The variables of interest are the output current i s (both positive and negative sequence) and the internal dynamics, which contains the circulating current i c , energy sum of each phase W Σ , energy difference W ∆ , and sum capacitor voltage.

Figure 2 shows internal control results when traditional control is used. It is obvious that, once the voltage is unbalanced (0.7 s), the control is no longer functional. As a result, oscillations in the three-phase circulating current (second plot) and the sum of capacitors voltages (first plot) are increasing in amplitude and this leads to instability. Control of the energy sum W Σ (third plot) is slow and has some steady-state error. Meanwhile, the control of energy difference has a lot of variance from zero reference. Results of internal control through the proposed method are shown in Figure 3 (same variables as before). It is noticed that, after the fault occurs, the closed-loop behavior is stabilized. Two of the circulating currents stabilize at about the same value, while that of the third phase stabilizes at a larger steady-state value.

Figure 4 shows the grid negative-sequence current i - s when conventional control is used. The result shows the closedloop performance of the grid current negative sequence d and q axis components, while in the case of GA-optimized state-feedback control it is obvious that in Figure 5 the proposed controller improves the negative sequence grid reference tracking in terms of accuracy and transients between faulty and normal grid circumstance, which has a good influence on power evolution. Note that, only the negative sequence current is shown, while both controllers have similar effects on the positive sequence.

In Figures 6 and7 the results for the circulating current i c and energy difference W ∆ for phase c are introduced (similar results in the other two phases). This result indicates that the proposed GA performance is comparable to that of the standard conventional control, and even better when the fault occurs, the cascaded control tending to be unstable in such condition.

Simulation-Based Robustness Checking

To perform the simulation-based robustness checking, a parameter study is performed, that is, it is supposed that some parameters may vary in some ranges. The uncertainty system parameters has also been investigated for the same test scenario. For instance, the sub-module capacitance C, the arm's inductance L, and the arm's resistance R were let to vary randomly within +/-20%, around their nominal values. The performance of the proposed GA optimized state-feedback controller being tested under these conditions Figure 8 for the circulating current i c . As can be seen, the controller's dynamics response is still pretty good. The controller obtained with a parameter inaccuracy converges to the desired reference but with some higher ripples. This result suggests that the circulating current using the proposed controller converges to a boundary layer near the specified reference. A systematic robustness study should confirm this result.

CONCLUSION

In this paper, a GA-optimized MIMO state-feedback control of both the grid and the circulating current in MMCs has been proposed. This approach can be seen as an alternative to the LQR and pole placement methods. Although the proposed approach requires some computation time to identify the "optimal" poles, this will be done just once and in an off-line way. Furthermore, there are very few parameters to tune which can save a lot of design time. Under imbalanced grid configurations, simulations reveal improved performance as compared to the traditional control. Future work should focus on developing a control method that takes explicitly account of positive, negative, and zero sequences.

Fig. 1 .

 1 Fig. 1. Structure of a typical three-phase MMC, showingall the important variables including the dc and ac bus[START_REF] Sharifabadi | Design, Control and Application of Modular Multilevel Converters for HVDC Transmission Systems[END_REF]

Fig. 2 .

 2 Fig. 2. Simulation results obtained for the cascaded conventional internal control of both i c and i s , under unbalanced grid conditions starting at time 0.7 s and lasting for 0.4 s

Fig. 6 .

 6 Fig. 6. Phase c circulating current i c control: comparison between conventional and GA-optimized statefeedback control under unbalanced grid conditions