Existence and characterization of regions minimizing perimeter under a volume constraint inside Euclidean cones

Ritoré, Manuel; Rosales, César

doi:10.1090/S0002-9947-04-03537-8

Existence and characterization of regions minimizing perimeter under a volume constraint inside Euclidean cones
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by Manuel Ritoré and César Rosales

Trans. Amer. Math. Soc. 356 (2004), 4601-4622

DOI: https://doi.org/10.1090/S0002-9947-04-03537-8

Published electronically: April 27, 2004

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Abstract:

We study the problem of existence of regions separating a given amount of volume with the least possible perimeter inside a Euclidean cone. Our main result shows that nonexistence for a given volume implies that the isoperimetric profile of the cone coincides with the one of the half-space. This allows us to give some criteria ensuring existence of isoperimetric regions: for instance, local convexity of the cone at some boundary point. We also characterize which are the stable regions in a convex cone, i.e., second order minima of perimeter under a volume constraint. From this it follows that the isoperimetric regions in a convex cone are the euclidean balls centered at the vertex intersected with the cone.

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