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Quantitative Stability for Minkowski's problem

Károly Böröczky, João Miguel Machado, João P. G. Ramos

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 $μ,ν$ satisfying a quantitative form of the classical dispersion assumptions yielding existence of such bodies, we have a control of the form \[ α(E_μ, E_ν)^2 \le C \mathrm{d}_{\mathrm{C}}(μ,ν)^{1 + \frac{1}{n}}, \] where $α$ 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.

Quantitative Stability for Minkowski's problem

Abstract

We derive quantitative stability results for Minkowski bodies, as well as their counterparts, the -Minkowski bodies in the range . We prove that, for every pair of probability measures satisfying a quantitative form of the classical dispersion assumptions yielding existence of such bodies, we have a control of the form where denotes the Fraenkel asymmetry and 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.
Paper Structure (9 sections, 12 theorems, 132 equations, 1 figure)

This paper contains 9 sections, 12 theorems, 132 equations, 1 figure.

Table of Contents

  1. Overview on Minkowski problems
  2. Preliminaries on convex bodies and Minkowski problems
  3. Inner and circumradii estimates with $\Theta(S_K)$
  4. On the $L_p$-Minkowski problem for $p>1$
  5. Quantitative stability for $p=1$
  6. Proof of Theorem \ref{['thm.quantitative_stability_p=1']}
  7. Quantitative stability for $1 < p \neq n$
  8. Quantitative versions of $L_p$ Brunn-Minkowski and isoperimetric inequalities
  9. Proof of Theorem \ref{['thm.quantitative_stability_p']}

Key Result

Theorem 1.1

Let $\mu, \nu \in \mathscr{P}(\mathbb{S}^{n-1})$ be two measures such that $\Theta(\mu), \Theta(\nu) \ge \vartheta >0$, and let $E_\mu, E_\nu$ be the associated Minkowski bodies centered at the origin. Then there is a constant $C_{\vartheta,n} >0$ depending only on $\vartheta$ and the dimension $n$

Figures (1)

  • Figure 1: Family of ellipsoids collapsing to a line as $\Theta$ goes to $0$.

Theorems & Definitions (25)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Remark 2.5
  • ...and 15 more