On the Converse of Prékopa's Theorem and Berndtsson's Theorem
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910791956955136 |
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| author | Xu, Wang Yang, Hui |
| author_facet | Xu, Wang Yang, Hui |
| contents | Given a continuous function $ϕ$ defined on a domain $Ω\subset\mathbb{R}^m\times\mathbb{R}^n$, we show that if a Prékopa-type result holds for $ϕ+ψ$ for any non-negative convex function $ψ$ on $Ω$, then $ϕ$ must be a convex function. Additionally, if the projection of $Ω$ onto $\mathbb{R}^m$ is convex, then $\overlineΩ$ is also convex. This provides a converse of Prékopa's theorem from convex analysis. We also establish analogous results for Berndtsson's theorem on the plurisubharmonic variation of Bergman kernels, showing that the plurisubharmonicity of weight functions and the pseudoconvexity of domains are necessary conditions in some sense. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_11456 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Converse of Prékopa's Theorem and Berndtsson's Theorem Xu, Wang Yang, Hui Complex Variables 32U05, 32T99, 32A36, 26B25, 52A20 Given a continuous function $ϕ$ defined on a domain $Ω\subset\mathbb{R}^m\times\mathbb{R}^n$, we show that if a Prékopa-type result holds for $ϕ+ψ$ for any non-negative convex function $ψ$ on $Ω$, then $ϕ$ must be a convex function. Additionally, if the projection of $Ω$ onto $\mathbb{R}^m$ is convex, then $\overlineΩ$ is also convex. This provides a converse of Prékopa's theorem from convex analysis. We also establish analogous results for Berndtsson's theorem on the plurisubharmonic variation of Bergman kernels, showing that the plurisubharmonicity of weight functions and the pseudoconvexity of domains are necessary conditions in some sense. |
| title | On the Converse of Prékopa's Theorem and Berndtsson's Theorem |
| topic | Complex Variables 32U05, 32T99, 32A36, 26B25, 52A20 |
| url | https://arxiv.org/abs/2501.11456 |