Isoperimetric inequalities in Euclidean convex bodies

Ritoré, Manuel; Vernadakis, Efstratios

doi:10.1090/S0002-9947-2015-06197-2

Isoperimetric inequalities in Euclidean convex bodies
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by Manuel Ritoré and Efstratios Vernadakis PDF

Trans. Amer. Math. Soc. 367 (2015), 4983-5014 Request permission

Abstract:

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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