Dans cet article nous étudions le problème d’optimisation de forme : trouver le domaine plan d’aire minimale parmi les convexes de largeur constante et d’inradius donnés. Dans la littérature, ce problème est attribué à Bonnesen qui a proposé une conjecture pour le domaine optimal. Nous donnons ici une réponse complète à ce problème en décrivant les domaines optimaux pour tout choix de la largeur et de l’inradius. Ces domaines sont des polygones de Reuleaux particuliers.
In this paper we address the following shape optimization problem: find the planar domain of least area, among the sets with prescribed constant width and inradius. In the literature, the problem is ascribed to Bonnesen, who proposed it in [3]. In the present work, we give a complete answer to the problem, providing an explicit characterization of optimal sets for every choice of width and inradius. These optimal sets are particular Reuleaux polygons.
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@article{JEP_2021__8__415_0,
author = {Antoine Henrot and Ilaria Lucardesi},
title = {Body of constant width with minimal area in a given annulus},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {415--438},
publisher = {\'Ecole polytechnique},
volume = {8},
year = {2021},
doi = {10.5802/jep.150},
mrnumber = {4218163},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.150/}
}
TY - JOUR AU - Antoine Henrot AU - Ilaria Lucardesi TI - Body of constant width with minimal area in a given annulus JO - Journal de l’École polytechnique — Mathématiques PY - 2021 SP - 415 EP - 438 VL - 8 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.150/ DO - 10.5802/jep.150 LA - en ID - JEP_2021__8__415_0 ER -
%0 Journal Article %A Antoine Henrot %A Ilaria Lucardesi %T Body of constant width with minimal area in a given annulus %J Journal de l’École polytechnique — Mathématiques %D 2021 %P 415-438 %V 8 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.150/ %R 10.5802/jep.150 %G en %F JEP_2021__8__415_0
Antoine Henrot; Ilaria Lucardesi. Body of constant width with minimal area in a given annulus. Journal de l’École polytechnique — Mathématiques, Tome 8 (2021), pp. 415-438. doi : 10.5802/jep.150. https://jep.centre-mersenne.org/articles/10.5802/jep.150/
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