Quantitative Stability for Minkowski's problem

TL;DR

This paper establishes quantitative stability estimates for Minkowski and Lp-Minkowski bodies, relating measure distances to geometric deviations with sharp exponents in certain cases.

Contribution

It provides new stability bounds for Minkowski bodies using a variational approach and sharp exponents, especially in the Hausdorff distance and Fraenkel asymmetry.

Findings

Hausdorff distance stability exponent is sharp.

Fraenkel asymmetry stability exponent is optimal in dimension 2.

Derived bounds relate measure differences to geometric deviations.

Abstract

We derive quantitative stability results for Minkowski bodies, as well as their counterparts, the $L_p$-Minkowski bodies in the range $1 \le p \neq n$. We prove that, for every pair of probability measures $\mu,\nu$ satisfying a quantitative form of the classical dispersion assumptions yielding existence of such bodies, we have a control of the form \[ \inf_{x\in \mathbb{R}^n}\mathrm{d_H}(E_\mu, x + E_\nu) \le C \mathrm{d_C}(\mu,\nu)^{\frac{1}{n-1}}, \quad \alpha(E_\mu, E_\nu)^2 \le C \mathrm{d_C}(\mu,\nu)^{1 + \frac{1}{n-1}}, \] where $\mathrm{d_H}$ denotes the Hausdorff distance, $\alpha$ denotes the Fraenkel asymmetry and $\mathrm{d_C}$ is the dual-convex distance of probability measures on the sphere. Our arguments are based on a variational problem whose optimizers are Minkowski bodies, for which we can obtain strong-concavity properties with the quantitative Brunn-Minkowski and isoperimetric inequalities. While the exponent in the Hausdorff distance is sharp, the exponent in the Fraenkel asymmetry is optimal in dimension $2$.