Quantitative Stability for Minkowski's problem

Abstract

We derive quantitative stability results for Minkowski bodies, as well as their counterparts, the $L_p$-Minkowski bodies in the range $1\le p\ne n$. We prove that, for every pair of probability measures $\hat{I}¼,\hat{I}½$ satisfying a quantitative form of the classical dispersion assumptions yielding existence of such bodies, we have a control of the form \[ \inf_{x\in \mathbb{R}^n}\mathrm{d_H}(E_μ, x + E_ν) \le C \mathrm{d_C}(μ,ν)^{\frac{1}{n-1}}, \quad α(E_μ, E_ν)^2 \le C \mathrm{d_C}(μ,ν)^{1 + \frac{1}{n-1}}, \] where ${\mathrm{d}}_{\mathrm{H}}$ denotes the Hausdorff distance, $\hat{I}\pm$ denotes the Fraenkel asymmetry and ${\mathrm{d}}_{\mathrm{C}}$ 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 $2$.