Objets convexes de largeur constante (en 2D) ou d'épaisseur constante (en 3D) : du neuf avec du vieux

International audienceLes objets convexes de largeur constante (dans le plan) ou d'épaisseur constante (dans l'espace) ont fait l'objet d'une attention soutenue de la part des mathématiciens du XIXe comme du XXe siècle, y compris par les plus célèbres d'entre eux (H. Minkowski, H. Lebesgue, W. Blaschke, A. Hurwitz, etc.). Malgré tous les efforts déployés et le nombre de résultats obtenus, certains problèmes posés depuis longtemps à propos de ces objets convexes restent encore ouverts. Les techniques modernes comme celles issues du calcul variationnel ou du contrôle optimal ont néanmoins permis soit de retrouver d'une nouvelle manière des résultats déjà démontrés, soit d'en améliorer significativement certains autres. Dans cet article, qui se veut de synthèse et à but essentiellement pédagogique, nous passons en revue les propriétés et caractérisations essentielles, plutôt de type " variationnel ", des corps convexes de largeur constante (en 2D) ou d'épaisseur constante (en 3D), en insistant sur les différences fondamentales en 2D ou 3D ; ce faisant, nous arrivons sur le front de la recherche récente sur les problèmes restés ouverts, en particulier la conjecture sur le corps convexe de l'espace d'épaisseur constante donnée et de volume minimal

Sensitivity of the “intermediate point” in the mean value theorem: an approach via the Legendre-Fenchel transformation

We study the sensitivity, essentially the differentiability, of the so-called “intermediate point” c in the classical mean value theorem fa-f(b)b-a=f'(c)$\frac{f(a)-f(b)}{b-a}={f}^{\prime}(c)$we provide the expression of its gradient ∇c(d,d), thus giving the asymptotic behavior of c(a, b) when both a and b tend to the same point d. Under appropriate mild conditions on f, this result is “universal” in the sense that it does not depend on the point d or the function f. The key tool to get at this result turns out to be the Legendre-Fenchel transformation for convex functions

Polytopal balls arising in optimization

We study a family of polytopes and their duals, that appear in various optimization problems as the unit balls for certain norms. These two families interpolate between the hypercube, the unit ball for the $\infty$-norm, and its dual cross-polytope, the unit ball for the $1$-norm. We give combinatorial and geometric properties of both families of polytopes such as their $f$-vector, their volume, and the volume of their boundary.Comment: 16 pages, 2 figure

A characterization by optimization of the Monge point of a tetrahedron

Abstract. "... nihil omnino in mundo contingint, in quo non maximi minimive ratio quapiam eluceat", translated into "... nothing in all the world will occur in which no maximum or minimum rule is somehow shining forth", used to say L.Euler in 1744. This is confirmed by numerous applications of mathematics in physics, mechanics, economy, etc. In this note, we show that it is also the case for the classical "centers" of a tetrahedron, more specifically for the so-called Monge point (the substitute of the notion of orthocenter for a tetrahedron). To the best of our knowledge, the characterization of the Monge point of a tetrahedron by optimization, that we are going to present, is new. To begin with... What kind of tetrahedron? Let T = ABCD be a tetrahedron in the three dimensional space R 3 (equipped with the usual Euclidean and affine structures); the points A, B, C, D are supposed not to lie in a plane, of course. We begin with two particular types of tetrahedra and, then, with increase in generality, we can classify the tetrahedra into several classes. Here they are: -The regular tetrahedron. This tetrahedron enjoys so many symmetries that it is not very interesting from the optimization viewpoint: all the "centers" usually associated with a tetrahedron (and that we are going to visit again in the next paragraph) coincide. -The trirectangular tetrahedra. They are generalizations to the space of rectangular triangles in the plane. A trirectangular tetrahedron OABC has (two by two) three perpendicular faces OBC, OAB, OAC and a "hypothenuse-face" ABC; such a tetrahedron enjoys a remarkable relationship between areas of its faces (see -The orthocentric tetrahedra. Curiously enough, the four altitudes of a tetrahedron generally do not meet at a point; when this happens, the tetrahedron is called orthocentric. A common characterization of orthocentric tetrahedra is as follows: a tetrahedron is orthocentric if and only if the opposite edges (two by two) are orthogonal. This class of tetrahedra is by far the most studied one in the literature. Regular and trirectangular tetrahedra are indeed orthocentric. -General tetrahedra. Like for triangles, three specific "centers" can be defined for any tetrahedron: the centroid or isobarycenter, the incenter and the circumcenter. We shall see their characterization by optimization, as for some other points, in the next section. As said before, the altitudes do not necessarily meet at a point; moreover, the projection of any vertex on the opposite face does not necessarily coincide with the orthocenter of this face. The notion of orthocenter will be held by a new point: the so-called Monge point

A Fresh Variational-Analysis Look at the Positive Semidefinite Matrices World

International audienceEngineering sciences and applications of mathematics show unambiguously that positive semidefiniteness of matrices is the most important generalization of non-negative real num- bers. This notion of non-negativity for matrices has been well-studied in the literature; it has been the subject of review papers and entire chapters of books. This paper reviews some of the nice, useful properties of positive (semi)definite matrices, and insists in particular on (i) characterizations of positive (semi)definiteness and (ii) the geometrical properties of the set of positive semidefinite matrices. Some properties that turn out to be less well-known have here a special treatment. The use of these properties in optimization, as well as various references to applications, are spread all the way through. The "raison d'être" of this paper is essentially pedagogical; it adopts the viewpoint of variational analysis, shedding new light on the topic. Important, fruitful, and subtle, the positive semidefinite world is a good place to start with this domain of applied mathematics

Le théorème déterminantal en bref

International audienceDans cette note à caractère essentiellement pédagogique, nous présentons le théorème déterminantal de CRAIG-SAKAMOTO, accompagné de notre démonstration favorite. Son histoire et une de ses applications sont brossées à grands traits. Nous concluons en suggérant une nouvelle piste de démonstration, que nous conjecturons être possible mais que nous n’avons pu mener à bien

Pierre Fermat: člověk, doba, velké výsledky

International audienceJ. B. Hiriart-Urruty v tomto článku představuje osobu Pierra de Fermata a jeho dobu, dále pak výsledky jeho celoživotní práce v různých vědeckých oblastech, podrobněji pak o Velké Fermatově větě, která se dočkala svého důkazu až v roce 1993. Autor dodává, že není snadné najít vhodnou literaturu pojednávající o Fermatovi