The Gaussian Minkowski problem for epigraphs of convex functions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912539868135424 |
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| author | Li, Xiao Ye, Deping |
| author_facet | Li, Xiao Ye, Deping |
| contents | A variational formula is derived by combining the Gaussian volume of the epigraph of a convex function $φ$ and the perturbation of $φ$ via the infimal convolution. This formula naturally leads to a Borel measure on $\mathbb{R}^n$ and a Borel measure on the unit sphere $S^{n-1}$. The resulting Borel measure on $\mathbb{R}^n$ will be called the Euclidean Gaussian moment measure of the convex function $φ$, and the related Minkowski-type problem will be studied. In particular, the newly posed Minkowski problem is solved under some mild and natural conditions on the pre-given measure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_12028 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Gaussian Minkowski problem for epigraphs of convex functions Li, Xiao Ye, Deping Functional Analysis Analysis of PDEs Metric Geometry 26B25, 52A40, 52A41, 35G20 A variational formula is derived by combining the Gaussian volume of the epigraph of a convex function $φ$ and the perturbation of $φ$ via the infimal convolution. This formula naturally leads to a Borel measure on $\mathbb{R}^n$ and a Borel measure on the unit sphere $S^{n-1}$. The resulting Borel measure on $\mathbb{R}^n$ will be called the Euclidean Gaussian moment measure of the convex function $φ$, and the related Minkowski-type problem will be studied. In particular, the newly posed Minkowski problem is solved under some mild and natural conditions on the pre-given measure. |
| title | The Gaussian Minkowski problem for epigraphs of convex functions |
| topic | Functional Analysis Analysis of PDEs Metric Geometry 26B25, 52A40, 52A41, 35G20 |
| url | https://arxiv.org/abs/2508.12028 |