preprint

Inserted: 21 aug 2026
Last Updated: 21 aug 2026

Year: 2026

Abstract:

We study a shape optimization problem among convex bodies in $\mathbb{R}^n$ that minimize or maximize weighted perimeters of the form $\int_{\partialΩ} φ(
x
) \, \mathrm{d} H^{n-1}(x)$ 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 $\lvert{}x\rvert{}^p$ for $p \in (0,2]$ and $\lvert{}x\rvert{}^{-\alpha}$ for $\alpha\in (0,1)$, among convex curves satisfying a symmetry assumption.