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We study the problem of existence of regions separating a given amount ofvolume with the least possible perimeter inside a Euclidean cone. Our mainresult shows that nonexistence for a given volume implies that theisoperimetric profile of the cone coincides with the one of the half-space.This allows us to give some criteria ensuring existence of isoperimetricregions: 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 itfollows that the isoperimetric regions in a convex cone are the euclidean ballscentered at the vertex intersected with the cone.

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