Isoperimetric inequalities in convex cylinders and cylindrically bounded convex bodies

TL;DR

This paper investigates the isoperimetric profile of convex cylinders and cylindrically bounded convex sets, establishing properties like concavity, existence, and geometric descriptions of optimal regions for various volumes.

Contribution

It provides new results on the concavity, existence, and geometric characterization of isoperimetric regions in convex cylinders and bounded convex bodies.

Findings

Concavity of the isoperimetric profile established.

Existence of isoperimetric regions proven.

Geometric descriptions for small and large volume regions provided.

Abstract

In this paper we consider the isoperimetric profile of convex cylinders $K\times\mathbb{R}^q$, where $K$ is an $m$-dimensional convex body, and of cylindrically bounded convex sets, i.e, those with a relatively compact orthogonal projection over some hyperplane of $\mathbb{R}^{n+1}$, asymptotic to a right convex cylinder of the form $K\times\mathbb{R}$, with $K\subset\mathbb{R}^n$. Results concerning the concavity of the isoperimetric profile, existence of isoperimetric regions, and geometric descriptions of isoperimetric regions for small and large volumes are obtained.