Canal Classes and Cheeger Sets

TL;DR

This paper investigates the optimal volume-to-surface-area ratio of convex bodies sharing a fixed projection, characterizing when cylinders are optimal in dimension 3 using Cheeger sets, and exploring related inequalities for canal classes.

Contribution

It provides a characterization of when cylinders maximize the ratio in dimension 3 via Cheeger sets and extends partial results to higher dimensions, introducing canal classes.

Findings

Characterization of the optimal ratio in dimension 3 using Cheeger sets.

Partial results and inequalities for convex bodies in higher dimensions.

Introduction of canal classes satisfying related geometric inequalities.

Abstract

Giannopoulos, Hartzoulaki and Paouris asked in \cite{GHP} whether the best ratio between volume and surface area of convex bodies sharing a given orthogonal projection onto a fixed hyperplane is attained in the limit by a cylinder over the given projection. The answer to the question is known to be negative. In this paper, we prove a characterization of the positive answer in dimension $3$, using the Cheeger set of the common projection. A partial characterization is given in higher dimensions. We also prove that certain canal classes of convex bodies provide families of convex bodies satisfying a closely related inequality for a similar ratio.