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Isoperimetric inequalities in Euclidean convex bodies


Authors: Manuel Ritoré and Efstratios Vernadakis
Journal: Trans. Amer. Math. Soc. 367 (2015), 4983-5014
MSC (2010): Primary 49Q10, 49Q20, 52B60
DOI: https://doi.org/10.1090/S0002-9947-2015-06197-2
Published electronically: February 19, 2015
MathSciNet review: 3335407
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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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Additional Information

Manuel Ritoré
Affiliation: Departamento de Geometría y Topología, Universidad de Granada, E–18071 Granada, España
Email: ritore@ugr.es

Efstratios Vernadakis
Affiliation: Departamento de Geometría y Topología, Universidad de Granada, E–18071 Granada, España
Email: stratos@ugr.es

DOI: https://doi.org/10.1090/S0002-9947-2015-06197-2
Keywords: Isoperimetric inequalities, isoperimetric profile, convex bodies, polytopes, Hausdorff distance, Lipschitz distance, perimeter-minimizing sets
Received by editor(s): April 26, 2013
Published electronically: February 19, 2015
Additional Notes: Both authors have been supported by MICINN-FEDER grant MTM2010-21206-C02-01 and Junta de Andalucía grants FQM-325 and P09-FQM-5088
Dedicated: Dedicated to Carlos Benítez on his 70th birthday
Article copyright: © Copyright 2015 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.