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

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

2003-07-16

arxiv.org/abs/math/0307217v1

Mathematics

Differential Geometry

21 pages, no figures

Scientific paper

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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