Quantitative Stability for Minkowski's problem
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866913121574060032 |
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| author | Böröczky, Károly Machado, João Miguel Ramos, João P. G. |
| author_facet | Böröczky, Károly Machado, João Miguel Ramos, João P. G. |
| contents | 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 \[
\inf_{x\in \mathbb{R}^n}\mathrm{d_H}(E_μ, x + E_ν) \le C \mathrm{d_C}(μ,ν)^{\frac{1}{n-1}}, \quad
α(E_μ, E_ν)^2 \le C \mathrm{d_C}(μ,ν)^{1 + \frac{1}{n-1}}, \] where $\mathrm{d_H}$ denotes the Hausdorff distance, $α$ 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 $2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_17726 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Quantitative Stability for Minkowski's problem Böröczky, Károly Machado, João Miguel Ramos, João P. G. Analysis of PDEs Metric Geometry Optimization and Control 52A21, 52A40, 49Q10 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 \[ \inf_{x\in \mathbb{R}^n}\mathrm{d_H}(E_μ, x + E_ν) \le C \mathrm{d_C}(μ,ν)^{\frac{1}{n-1}}, \quad α(E_μ, E_ν)^2 \le C \mathrm{d_C}(μ,ν)^{1 + \frac{1}{n-1}}, \] where $\mathrm{d_H}$ denotes the Hausdorff distance, $α$ 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 $2$. |
| title | Quantitative Stability for Minkowski's problem |
| topic | Analysis of PDEs Metric Geometry Optimization and Control 52A21, 52A40, 49Q10 |
| url | https://arxiv.org/abs/2603.17726 |