[Le diagramme manquant]
Dans cet article, nous nous intéressons à des inégalités géométriques universelles “optimales” mettant en jeu l’aire, le diamètre et l’inradius des corps convexes. Le terme“optimal” doit être compris dans le sens suivant : nous étudions la question de la minimisation/maximisation de la mesure de Lebesgue d’un corps convexe parmi tous les ensembles convexes de diamètre et de rayon intérieur donnés. Le problème de minimisation dans le cas bidimensionnel a été résolu dans un article précédent, par M. Hernandez–Cifre et G. Salinas. Dans cet article, nous fournissons une généralisation au cas -dimensionnel basée sur une approche différente, ainsi que la résolution complète du problème de maximisation dans le cas bidimensionnel. Cela nous permet de déterminer complètement le diagramme de Blaschke–Santaló bidimensionnel pour les corps convexes planaires par rapport aux trois grandeurs aire, diamètre et inradius dans les espaces euclidiens, noté . Un tel diagramme est utilisé pour déterminer la gamme des valeurs possibles de l’aire des ensembles convexes en fonction de leur diamètre et de leur inradius. Bien que cette question de géométrie convexe semble assez élémentaire, elle n’avait pas trouvé de réponse jusqu’à présent. Ceci est probablement lié au fait que la description du diagramme utilise des ensembles convexes particuliers inattendus, tels qu’une sorte de nonagone lissé inscrit dans un triangle équilatéral.
In this paper we are interested in “optimal” universal geometric inequalities involving the area, diameter and inradius of convex bodies. The term “optimal” is to be understood in the following sense: we tackle the issue of minimizing/maximizing the Lebesgue measure of a convex body among all convex sets of given diameter and inradius. The minimization problem in the two-dimensional case has been solved in a previous work, by M. Hernandez-Cifre and G. Salinas. In this article, we provide a generalization to the -dimensional case based on a different approach, as well as the complete solving of the maximization problem in the two-dimensional case. This allows us to completely determine the so-called 2-dimensional Blaschke–Santaló diagram for planar convex bodies with respect to the three magnitudes area, diameter and inradius in euclidean spaces, denoted . Such a diagram is used to determine the range of possible values of the area of convex sets depending on their diameter and inradius. Although this question of convex geometry appears to be quite elementary, it had not been answered until now. This is probably related to the fact that the diagram description uses unexpected particular convex sets, such as a kind of smoothed nonagon inscribed in an equilateral triangle.
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Keywords: shape optimization, diameter, inradius, convex geometry, 2-cap bodies, Blaschke–Santaló diagram
Mot clés : optimisation de forme, diamètre, inradius, géométrie convexe, 2-cap bodies, diagramme de Blaschke–Santaló
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@article{AIF_2022__72_5_1941_0,
author = {Delyon, Alexandre and Henrot, Antoine and Privat, Yannick},
title = {The missing $\protect (A,D,r)$ diagram},
journal = {Annales de l'Institut Fourier},
pages = {1941--1992},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {72},
number = {5},
year = {2022},
doi = {10.5802/aif.3484},
language = {en},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3484/}
}
TY - JOUR AU - Delyon, Alexandre AU - Henrot, Antoine AU - Privat, Yannick TI - The missing $\protect (A,D,r)$ diagram JO - Annales de l'Institut Fourier PY - 2022 SP - 1941 EP - 1992 VL - 72 IS - 5 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3484/ DO - 10.5802/aif.3484 LA - en ID - AIF_2022__72_5_1941_0 ER -
%0 Journal Article %A Delyon, Alexandre %A Henrot, Antoine %A Privat, Yannick %T The missing $\protect (A,D,r)$ diagram %J Annales de l'Institut Fourier %D 2022 %P 1941-1992 %V 72 %N 5 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3484/ %R 10.5802/aif.3484 %G en %F AIF_2022__72_5_1941_0
Delyon, Alexandre; Henrot, Antoine; Privat, Yannick. The missing $\protect (A,D,r)$ diagram. Annales de l'Institut Fourier, Tome 72 (2022) no. 5, pp. 1941-1992. doi : 10.5802/aif.3484. https://aif.centre-mersenne.org/articles/10.5802/aif.3484/
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