Fixed and Periodic Points of the Intersection Body Operator
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866915335602438144 |
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| author | Milman, Emanuel Shabelman, Shahar Yehudayoff, Amir |
| author_facet | Milman, Emanuel Shabelman, Shahar Yehudayoff, Amir |
| contents | The intersection body $IK$ of a star-body $K$ in $\mathbb{R}^n$ was introduced by E. Lutwak following the work of H. Busemann, and plays a central role in the dual Brunn-Minkowski theory. We show that when $n \geq 3$, $I^2 K = c K$ iff $K$ is a centered ellipsoid, and hence $I K = c K$ iff $K$ is a centered Euclidean ball, answering long-standing questions by Lutwak, Gardner, and Fish-Nazarov-Ryabogin-Zvavitch. To this end, we recast the iterated intersection body equation as an Euler-Lagrange equation for a certain volume functional under radial perturbations, derive new formulas for the volume of $I K$, and introduce a continuous version of Steiner symmetrization for Lipschitz star-bodies, which (surprisingly) yields a useful radial perturbation exactly when $n\geq 3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_08171 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fixed and Periodic Points of the Intersection Body Operator Milman, Emanuel Shabelman, Shahar Yehudayoff, Amir Functional Analysis Metric Geometry The intersection body $IK$ of a star-body $K$ in $\mathbb{R}^n$ was introduced by E. Lutwak following the work of H. Busemann, and plays a central role in the dual Brunn-Minkowski theory. We show that when $n \geq 3$, $I^2 K = c K$ iff $K$ is a centered ellipsoid, and hence $I K = c K$ iff $K$ is a centered Euclidean ball, answering long-standing questions by Lutwak, Gardner, and Fish-Nazarov-Ryabogin-Zvavitch. To this end, we recast the iterated intersection body equation as an Euler-Lagrange equation for a certain volume functional under radial perturbations, derive new formulas for the volume of $I K$, and introduce a continuous version of Steiner symmetrization for Lipschitz star-bodies, which (surprisingly) yields a useful radial perturbation exactly when $n\geq 3$. |
| title | Fixed and Periodic Points of the Intersection Body Operator |
| topic | Functional Analysis Metric Geometry |
| url | https://arxiv.org/abs/2408.08171 |