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

Authors:
Manuel Ritoré and César Rosales

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

MSC (2000):
Primary 53C20, 49Q20

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

Published electronically:
April 27, 2004

MathSciNet review:
2067135

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Abstract | References | Similar Articles | Additional Information

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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Additional Information

**Manuel Ritoré**

Affiliation:
Departamento de Geometría y Topología, Universidad de Granada, E–18071 Granada, Spain

Email:
ritore@ugr.es

**César Rosales**

Affiliation:
Departamento de Geometría y Topología, Universidad de Granada, E–18071 Granada, Spain

Email:
crosales@ugr.es

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

Keywords:
Isoperimetric regions,
stability,
hypersurfaces with constant mean curvature

Received by editor(s):
March 6, 2003

Received by editor(s) in revised form:
July 22, 2003

Published electronically:
April 27, 2004

Additional Notes:
Both authors were supported by MCyT-Feder research project BFM2001-3489

Article copyright:
© Copyright 2004
American Mathematical Society