Calculus of Variations and Geometric Measure Theory

A. Henrot - I. Lucardesi

A Blaschke-Lebesgue theorem for the Cheeger constant

created by lucardesi on 14 Nov 2020
modified on 21 Apr 2023


Accepted Paper

Inserted: 14 nov 2020
Last Updated: 21 apr 2023

Journal: Communications in Contemporary Mathematics
Year: 2023


In this paper we prove a new extremal property of the Reuleaux triangle: it maximizes the Cheeger constant among all bodies of (same) constant width. The proof relies on a fine analysis of the optimality conditions satisfied by an optimal Reuleaux polygon together with an explicit upper bound for the inradius of the optimal domain. As a possible perspective, we conjecture that this maximal property of the Reuleaux triangle holds for the first eigenvalue of the $p$-Laplacian for any $p\in (1,+\infty)$ (the current paper covers the case $p=1$ whereas the case $p=+\infty$ was already known).