Fixed and periodic points of the intersection body operators of lower orders
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911309131415552 |
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| author | Lin, Cheng Xiong, Ge |
| author_facet | Lin, Cheng Xiong, Ge |
| contents | For the intersection body operator of lower order $I_iK$ of a star body $K$ in $\mathbb{R}^n$, $i\in\{1, 2,\ldots, n-2\}$, we prove that $I_i^2K = cK$ iff $K$ is an origin-symmetric ball, and hence $I_iK = cK$ iff $K$ is an origin-symmetric ball. Combining the recent breakthrough (case $i = n-1$) of Milman, Shabelman and Yehudayoff (Invent. Math., 241 (2025), 509-558), slight modifications of two long-standing questions 8.6 and 8.7 posed by R. Gardner (Page 302, Geometric Tomography, Cambridge University Press, 1995) are completely solved. As applications, we show that for the spherical Radon transform $\mathcal{R}$, a non-negative $ρ\in L^{\infty}(\mathcal{S}^{n-1})$ satisfies $\mathcal{R}(ρ^i) = cρ$ for some $c>0$ iff $ρ$ is constant. Also, the sharp Busemann intersection type inequalities are established. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_26381 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fixed and periodic points of the intersection body operators of lower orders Lin, Cheng Xiong, Ge Metric Geometry 52A40, 52A30, 52A38, 42B15 For the intersection body operator of lower order $I_iK$ of a star body $K$ in $\mathbb{R}^n$, $i\in\{1, 2,\ldots, n-2\}$, we prove that $I_i^2K = cK$ iff $K$ is an origin-symmetric ball, and hence $I_iK = cK$ iff $K$ is an origin-symmetric ball. Combining the recent breakthrough (case $i = n-1$) of Milman, Shabelman and Yehudayoff (Invent. Math., 241 (2025), 509-558), slight modifications of two long-standing questions 8.6 and 8.7 posed by R. Gardner (Page 302, Geometric Tomography, Cambridge University Press, 1995) are completely solved. As applications, we show that for the spherical Radon transform $\mathcal{R}$, a non-negative $ρ\in L^{\infty}(\mathcal{S}^{n-1})$ satisfies $\mathcal{R}(ρ^i) = cρ$ for some $c>0$ iff $ρ$ is constant. Also, the sharp Busemann intersection type inequalities are established. |
| title | Fixed and periodic points of the intersection body operators of lower orders |
| topic | Metric Geometry 52A40, 52A30, 52A38, 42B15 |
| url | https://arxiv.org/abs/2510.26381 |