John-type decompositions for affinely-optimal positions of convex bodies
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866910023885520896 |
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| author | Grundbacher, Florian Kobos, Tomasz |
| author_facet | Grundbacher, Florian Kobos, Tomasz |
| contents | Many classical problems in convex geometry can be cast as optimization problems under certain containment conditions. The arguably best-understood example is volume-maximization of convex bodies contained in other convex bodies, where the John decomposition describes$\unicode{x2014}$and in the Euclidean case fully characterizes$\unicode{x2014}$the optimal positions. For many other such problems, however, no general optimality conditions are known. To address this, we generalize an approach of O. B. Ader to obtain a John-type decomposition as a necessary condition for affinely-optimal containment chains, i.e., chains $r L_1 + c \subseteq K \subseteq R L_2 + d$ for convex bodies $K, L_1, L_2 \subseteq \mathbb{R}^n$, translation vectors $c,d \in \mathbb{R}^n$, and reals $r,R > 0$ such that the ratio $\frac{R}{r}$ cannot be decreased by linearly transforming $K$. We again obtain sufficiency for optimality when ellipsoids are involved, and show how optimality conditions for various problems follow from our result. Our main applications concern the Banach-Mazur distance, where we provide necessary optimality conditions in the general case and a full characterization in the Euclidean case. Finally, we derive several consequences of these optimality conditions related to the Banach-Mazur distance to the Euclidean ball. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_14847 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | John-type decompositions for affinely-optimal positions of convex bodies Grundbacher, Florian Kobos, Tomasz Metric Geometry Functional Analysis 52A40 (Primary) 46B07, 46B20, 52A21 (Secondary) Many classical problems in convex geometry can be cast as optimization problems under certain containment conditions. The arguably best-understood example is volume-maximization of convex bodies contained in other convex bodies, where the John decomposition describes$\unicode{x2014}$and in the Euclidean case fully characterizes$\unicode{x2014}$the optimal positions. For many other such problems, however, no general optimality conditions are known. To address this, we generalize an approach of O. B. Ader to obtain a John-type decomposition as a necessary condition for affinely-optimal containment chains, i.e., chains $r L_1 + c \subseteq K \subseteq R L_2 + d$ for convex bodies $K, L_1, L_2 \subseteq \mathbb{R}^n$, translation vectors $c,d \in \mathbb{R}^n$, and reals $r,R > 0$ such that the ratio $\frac{R}{r}$ cannot be decreased by linearly transforming $K$. We again obtain sufficiency for optimality when ellipsoids are involved, and show how optimality conditions for various problems follow from our result. Our main applications concern the Banach-Mazur distance, where we provide necessary optimality conditions in the general case and a full characterization in the Euclidean case. Finally, we derive several consequences of these optimality conditions related to the Banach-Mazur distance to the Euclidean ball. |
| title | John-type decompositions for affinely-optimal positions of convex bodies |
| topic | Metric Geometry Functional Analysis 52A40 (Primary) 46B07, 46B20, 52A21 (Secondary) |
| url | https://arxiv.org/abs/2602.14847 |