Isoperimetric inequalities in convex bodies

In this thesis we study isoperimetric inequalities in convex bodies. We have divided the Thesis into five chapters. The first chapter includes the introduction and preliminaries. In Chapter~\ref{ch:2} we deal only with compact convex bodies and 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 body. Amongst other results, we shall prove the equivalence between Hausdorff and 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. In Chapter~\ref{ch:3} we consider the isoperimetric profile of convex cylinders $K\times \rr^q$, 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 $ \rr^{n+1}$, asymptotic to a right convex cylinder of the form $K\times \rr$, with $K\subset R^n$. 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. In Chapter~\ref{ch:4} we consider the problem of minimizing the relative perimeter under a volume constraint in the interior of a conically bounded convex set, i.e., an unbounded convex body admitting an \emph{exterior} asymptotic cone. Results concerning existence of isoperimetric regions, the behavior of the isoperimetric profile for large volumes, and a characterization of isoperimetric regions of large volume in conically bounded convex sets of revolution is obtained. Finally In Chapter~\ref{ch:5}, given a compact Riemannian manifold $M$, we show that large isoperimetric regions in $M\times \rr^k$ are tubular neighborhoods of $M\times \{x\}$ with $x\in\rr^k$. \thispagestyle{empty}