Euclidean balls solve some isoperimetric problems with nonradial weights

In this note we present the solution of some isoperimetric problems in open convex cones of ${\mathbb{R}}^n$ in which perimeter and volume are measured with respect to certain nonradial weights. Surprisingly, Euclidean balls centered at the origin (intersected with the convex cone) minimize the isoperimetric quotient. Our result applies to all nonnegative homogeneous weights satisfying a concavity condition in the cone. When the weight is constant, the result was established by Lions and Pacella in 1990.