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 \[ \alpha(E_\mu, E_\nu)^2 \le C \mathrm{ d}_{ \mathrm{ C}}(\mu,\nu)^{ 1 + \frac{ 1}{ n}}, \] where $\alpha$ 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.
Categories: math.AP math.MG math.OC