In this paper we consider the problem of minimizing the relative perimeter under a volume constraint in the interior of a convex body, i.e., a compact convex set in Euclidean space with interior points. We shall not impose any regularity assumption on the boundary of the convex set. Amongst other results, we shall prove that Hausdorff convergence in the space of convex bodies implies Lipschitz convergence, the continuity of the isoperimetric profile with respect to the Hausdorff distance, and the convergence in Hausdorff distance of sequences of isoperimetric regions and their free boundaries. We shall also describe the behavior of the isoperimetric profile for small volume and the behavior of isoperimetric regions for small volume.

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Contents Chapter 1. Introduction 1.1. Historical background 1.2. Outline of contents Chapter 2. Convex bodies and finite perimeter sets 2.1. Convex bodies and local convergence in Hausdorff distance 2.2. Finite perimeter sets and isoperimetric profile Chapter 3. Unbounded convex bodies of uniform geometry 3.1. Asymptotic cylinders 3.2. Convex bodies of uniform geometry 3.3. Density estimates and a concentration lemma 3.4. Examples Chapter 4. A generalized existence result 4.1. Preliminary results 4.2. Proof of the main result Chapter 5. Concavity of the isoperimetric profile 5.1. Continuity of the isoperimetric profile 5.2. Approximation by smooth sets 5.3. Concavity of the isoperimetric profile Chapter 6. Sharp isoperimetric inequalities and isoperimetric rigidity 6.1. Convex bodies with non-degenerate asymptotic cone 6.2. The isoperimetric profile for small volumes 6.3. Isoperimetric rigidity Chapter 7. The isoperimetric dimension of an unbounded convex body 7.1. An asymptotic isoperimetric inequality 7.2. Estimates on the volume growth of balls 7.3. Examples Bibliography iii iv CONTENTS

We consider the problem of minimizing the relative perimeter under a volume constraint in an unbounded convex body C ⊂ R n C\subset \mathbb {R}^{n} , without assuming any further regularity on the boundary of C 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 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.

In this paper we consider the isoperimetric profile of convex cylinders K × 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 n+1 , asymptotic to a right convex cylinder of the form K × , with K ⊂ 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.