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

Abstract: We study a shape optimization problem among convex bodies in $\mathbb{R}<sup>n$ that minimize or maximize weighted perimeters of the form $$\int_{\partialΩ} φ(|x|) \, \mathrm{d} H<sup>{n-1}(x)$$ 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 $(-a,a)\times {0} \subset \mathbb{R}<sup>2$ is the optimizer for a wide family of weights, including $|x|<sup>p$ for $p \in (0,2]$ and $|x|<sup>{-α}$ for $α\in (0,1)$, among convex curves satisfying a symmetry assumption.