Quantitative Stability for Minkowski's problem

We derive quantitative stability results for Minkowski bodies, as well as their counterparts, the $L_p$-Minkowski bodies in the range $1\le p\mathrm{\ne }n$. 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 $$\underset{x\in {\mathbb{R}}^n}{\inf }{\mathrm{d}}_{\mathrm{H}}(E_{\mu },x+E_{\nu })\le C{\mathrm{d}}_{\mathrm{C}}(\mu ,\nu {)}^{\frac{1}{n-1}},\alpha (E_{\mu },E_{\nu }{)}^2\le C{\mathrm{d}}_{\mathrm{C}}(\mu ,\nu {)}^{1+\frac{1}{n-1}},$$ where ${\mathrm{d}}_{\mathrm{H}}$ denotes the Hausdorff distance, $\alpha$ 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 $2$.