A Blaschke-Lebesgue Theorem for the Cheeger constant

2020/11/14 by Henrot, Antoine, Lucardesi, Ilaria · 1 citation
#49Q10 #49Q12 #52A10 #52A38 #Analysis of PDEs (math.AP) #FOS: Mathematics #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.2011.07244

Abstract

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∈ (1,+∞) (the current paper covers the case p=1 whereas the case p=+∞ was already known).

Cited by

• Shape optimization under width constraint: the Cheeger constant and the torsional rigidity

Related

• On the Blaschke-Lebesgue theorem for the Cheeger constant via areas and\n perimeters of inner parallel sets
• The Blaschke-Lebesgue problem for constant width bodies of revolution
• A direct proof of a theorem of Blaschke and Lebesgue
• The Blaschke-Lebesgue theorem revisited
• A reverse isoperimetric inequality for the Cheeger constant under width constraint