# Shape optimization in the class of bodies of constant width.

## Ilaria Lucardesi (Institut Elie Cartan de Lorraine)

created by miranda on 01 Mar 2021

18 mar 2021 -- 15:00   [open in google calendar]

Dipartimento di Matematica e Informatica di Ferrara (on-line)

Abstract.

A body of constant width $d$ is a convex set with the following property: every pair of parallel (distinct) planes tangent to it have distance $d$ from each other, independently of their direction. This property is of course enjoyed by the ball, but it is shared by many other sets. One of the first shape optimization results in this class dates back to the late 1800s: Barbier, in 1860, showed that all plane convex sets of constant width $d$ have the same perimeter, equal to $\pi d$. An immediate consequence, by the classical isoperimetric inequality, is that the disk maximizes the area in the class. Some years after Barbier, at the beginning of the 1900s, Blaschke and Lebesgue proved that the set which minimizes the area is the so-called Reuleaux triangle, whose boundary is made of three arcs of circle of radius $d$ with same length (equal to $\pi d/3$). Since then, many alternative proofs of this result, with very different flavors, appeared (e.g., Eggleston '52, Besicovitch '63, Campi-Colesanti-Gronchi '96, Ghandehari '96, Harrell '02), and several other shape functionals have been considered, of geometric and spectral type. In the first part of the talk, I will introduce some tools to represent planar bodies of constant width, both analytically and numerically. In the second part, I will present a recent result, obtained in collaboration with Antoine Henrot, about the maximization of the Cheeger constant among the planar bodies of prescribed constant width.

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