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 $\int_{\partial\Omega} \phi(|x|) \, \mathrm{d} H^{n-1}(x)$ 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.