Continuity and monotonicity of weighted perimeters of convex bodies

We investigate weighted perimeters of convex bodies in ${\mathbb{R}}^n$. Our main result states that the weighted perimeter is continuous with respect to the Hausdorff metric for every radial weight function. We also establish monotonicity under inclusion under suitable assumptions on the radial weight, and show that these assumptions are sharp. These results are applied to show the existence of minimizers for a shape optimization problem.