{"title":"对数凹函数的里兹α元能及相关闵科夫斯基问题","authors":"Niufa Fang, Deping Ye, Zengle Zhang","doi":"arxiv-2408.16141","DOIUrl":null,"url":null,"abstract":"We calculate the first order variation of the Riesz $\\alpha$-energy of a\nlog-concave function $f$ with respect to the Asplund sum. Such a variational\nformula induces the Riesz $\\alpha$-energy measure of log-concave function $f$,\nwhich will be denoted by $\\mathfrak{R}_{\\alpha}(f, \\cdot)$. We pose the related\nRiesz $\\alpha$-energy Minkowski problem aiming to find necessary and/or\nsufficient conditions on a pregiven Borel measure $\\mu$ defined on $\\Rn$ so\nthat $\\mu=\\mathfrak{R}_{\\alpha}(f,\\cdot)$ for some log-concave function $f$.\nAssuming enough smoothness, the Riesz $\\alpha$-energy Minkowski problem reduces\nto a new Monge-Amp\\`{e}re type equation involving the Riesz $\\alpha$-potential.\nMoreover, this new Minkowski problem can be viewed as a functional counterpart\nof the recent Minkowski problem for the chord measures in integral geometry\nposed by Lutwak, Xi, Yang and Zhang (Comm.\\ Pure\\ Appl.\\ Math.,\\ 2024). The\nRiesz $\\alpha$-energy Minkowski problem will be solved under certain mild\nconditions on $\\mu$.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The Riesz $α$-energy of log-concave functions and related Minkowski problem\",\"authors\":\"Niufa Fang, Deping Ye, Zengle Zhang\",\"doi\":\"arxiv-2408.16141\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We calculate the first order variation of the Riesz $\\\\alpha$-energy of a\\nlog-concave function $f$ with respect to the Asplund sum. Such a variational\\nformula induces the Riesz $\\\\alpha$-energy measure of log-concave function $f$,\\nwhich will be denoted by $\\\\mathfrak{R}_{\\\\alpha}(f, \\\\cdot)$. We pose the related\\nRiesz $\\\\alpha$-energy Minkowski problem aiming to find necessary and/or\\nsufficient conditions on a pregiven Borel measure $\\\\mu$ defined on $\\\\Rn$ so\\nthat $\\\\mu=\\\\mathfrak{R}_{\\\\alpha}(f,\\\\cdot)$ for some log-concave function $f$.\\nAssuming enough smoothness, the Riesz $\\\\alpha$-energy Minkowski problem reduces\\nto a new Monge-Amp\\\\`{e}re type equation involving the Riesz $\\\\alpha$-potential.\\nMoreover, this new Minkowski problem can be viewed as a functional counterpart\\nof the recent Minkowski problem for the chord measures in integral geometry\\nposed by Lutwak, Xi, Yang and Zhang (Comm.\\\\ Pure\\\\ Appl.\\\\ Math.,\\\\ 2024). The\\nRiesz $\\\\alpha$-energy Minkowski problem will be solved under certain mild\\nconditions on $\\\\mu$.\",\"PeriodicalId\":501444,\"journal\":{\"name\":\"arXiv - MATH - Metric Geometry\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-08-28\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Metric Geometry\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2408.16141\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Metric Geometry","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.16141","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
The Riesz $α$-energy of log-concave functions and related Minkowski problem
We calculate the first order variation of the Riesz $\alpha$-energy of a
log-concave function $f$ with respect to the Asplund sum. Such a variational
formula induces the Riesz $\alpha$-energy measure of log-concave function $f$,
which will be denoted by $\mathfrak{R}_{\alpha}(f, \cdot)$. We pose the related
Riesz $\alpha$-energy Minkowski problem aiming to find necessary and/or
sufficient conditions on a pregiven Borel measure $\mu$ defined on $\Rn$ so
that $\mu=\mathfrak{R}_{\alpha}(f,\cdot)$ for some log-concave function $f$.
Assuming enough smoothness, the Riesz $\alpha$-energy Minkowski problem reduces
to a new Monge-Amp\`{e}re type equation involving the Riesz $\alpha$-potential.
Moreover, this new Minkowski problem can be viewed as a functional counterpart
of the recent Minkowski problem for the chord measures in integral geometry
posed by Lutwak, Xi, Yang and Zhang (Comm.\ Pure\ Appl.\ Math.,\ 2024). The
Riesz $\alpha$-energy Minkowski problem will be solved under certain mild
conditions on $\mu$.