Calculus of Variations and Geometric Measure Theory

G. P. Leonardi - M. Ritoré - E. Vernadakis

Isoperimetric inequalities in unbounded convex bodies

created by leonardi on 13 Jun 2016
modified on 21 Jan 2019


Accepted Paper

Inserted: 13 jun 2016
Last Updated: 21 jan 2019

Journal: Mem. AMS
Year: 2018


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 \textit{uniform geometry}. We then provide a handy characterization of the uniform geometry property and, by exploiting the notion of \textit{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 \textit{isoperimetric dimension}.