A Santaló inequality for the $L^p$-polar body
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914646203564032 |
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| author | Mastrantonis, Vlassis |
| author_facet | Mastrantonis, Vlassis |
| contents | In recent work with Berndtsson and Rubinstein, a notion of $L^p$-polarity was introduced, with classical polarity recovered in the limit $p\to\infty$, and $L^1$-polarity closely related to Bergman kernels of tube domains. A Santaló inequality for the $L^p$-polar was proved for symmetric convex bodies. The aim of this article is to remove the symmetry assumption. Thus, an $L^p$-Santaló inequality holds for any convex body after translation by the $L^p$-Santaló point. As a corollary, this yields an optimal upper bound on Bergman kernels of tube domains. The proof is by Steiner symmetrization, but unlike the symmetric case, a careful translation of the body is required before each symmetrization. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_10836 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Santaló inequality for the $L^p$-polar body Mastrantonis, Vlassis Functional Analysis Complex Variables In recent work with Berndtsson and Rubinstein, a notion of $L^p$-polarity was introduced, with classical polarity recovered in the limit $p\to\infty$, and $L^1$-polarity closely related to Bergman kernels of tube domains. A Santaló inequality for the $L^p$-polar was proved for symmetric convex bodies. The aim of this article is to remove the symmetry assumption. Thus, an $L^p$-Santaló inequality holds for any convex body after translation by the $L^p$-Santaló point. As a corollary, this yields an optimal upper bound on Bergman kernels of tube domains. The proof is by Steiner symmetrization, but unlike the symmetric case, a careful translation of the body is required before each symmetrization. |
| title | A Santaló inequality for the $L^p$-polar body |
| topic | Functional Analysis Complex Variables |
| url | https://arxiv.org/abs/2401.10836 |