Isoperimetric inequalities in unbounded convex bodies

TL;DR

This paper investigates the isoperimetric problem in unbounded convex bodies, introducing the concept of uniform geometry, proving existence and properties of isoperimetric regions, and analyzing the asymptotic behavior of the isoperimetric profile.

Contribution

It introduces the notion of uniform geometry for unbounded convex bodies and establishes existence, concavity, and connectedness properties of isoperimetric regions without boundary regularity assumptions.

Findings

Existence of isoperimetric regions in unbounded convex bodies with uniform geometry.

Strict concavity of the isoperimetric profile.

Connectedness of generalized isoperimetric regions.

Abstract

We consider the problem of minimizing the relative perimeter under a volume constraint in an unbounded convex body $C\subset \mathbb{R}^{n+1}$, without assuming any further regularity on the boundary of $C$. Motivated by an example of an unbounded convex body with null isoperimetric profile, we introduce the concept of unbounded convex body with uniform geometry. We then provide a handy characterization of the uniform geometry property and, by exploiting the notion of asymptotic cylinder of $C$, we prove existence of isoperimetric regions in a generalized sense. By an approximation argument we show the strict concavity of the isoperimetric profile and, consequently, the connectedness of generalized isoperimetric regions. We also focus on the cases of small as well as of large volumes; in particular we show existence of isoperimetric regions with sufficiently large volumes, for special classes of unbounded convex bodies. We finally address some questions about isoperimetric rigidity and analyze the asymptotic behavior of the isoperimetric profile in connection with the notion of isoperimetric dimension.