  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.
