Weighted Perimeters and pth Moments of Inertia of Convex Curves and Surfaces

Abstract

We study a shape optimization problem among convex bodies in ${\mathbb{R}}^n$ that minimize or maximize weighted perimeters of the form ${\int }_{\partial \Omega }\varphi (|x|)\mathrm{d}{H}^{n-1}(x)$ under a standard perimeter constraint. We prove the existence of extremals for general weight functions in any dimension. In dimension two, we prove that the degenerate needle configuration $(-a,a)\times \{0\}\subset {\mathbb{R}}^2$ is the optimizer for a wide family of weights, including $|x{|}^p$ for $p\in (0,2]$ and $|x{|}^{-\alpha }$ for $\alpha \in (0,1)$, among convex curves satisfying a symmetry assumption.