{"title":"快速估算多面体的体积","authors":"Alexander Barvinok, Mark Rudelson","doi":"10.1007/s11856-024-2615-z","DOIUrl":null,"url":null,"abstract":"<p>Let <i>P</i> be a bounded polyhedron defined as the intersection of the non-negative orthant ℝ<span>\n<sup><i>n</i></sup><sub>+</sub>\n</span> and an affine subspace of codimension <i>m</i> in ℝ<sup><i>n</i></sup>. We show that a simple and computationally efficient formula approximates the volume of <i>P</i> within a factor of <i>γ</i><sup><i>m</i></sup>, where <i>γ</i> > 0 is an absolute constant. The formula provides the best known estimate for the volume of transportation polytopes from a wide family.</p>","PeriodicalId":14661,"journal":{"name":"Israel Journal of Mathematics","volume":"937 1","pages":""},"PeriodicalIF":0.8000,"publicationDate":"2024-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A quick estimate for the volume of a polyhedron\",\"authors\":\"Alexander Barvinok, Mark Rudelson\",\"doi\":\"10.1007/s11856-024-2615-z\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Let <i>P</i> be a bounded polyhedron defined as the intersection of the non-negative orthant ℝ<span>\\n<sup><i>n</i></sup><sub>+</sub>\\n</span> and an affine subspace of codimension <i>m</i> in ℝ<sup><i>n</i></sup>. We show that a simple and computationally efficient formula approximates the volume of <i>P</i> within a factor of <i>γ</i><sup><i>m</i></sup>, where <i>γ</i> > 0 is an absolute constant. The formula provides the best known estimate for the volume of transportation polytopes from a wide family.</p>\",\"PeriodicalId\":14661,\"journal\":{\"name\":\"Israel Journal of Mathematics\",\"volume\":\"937 1\",\"pages\":\"\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2024-04-24\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Israel Journal of Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s11856-024-2615-z\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Israel Journal of Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s11856-024-2615-z","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
假设 P 是一个有界多面体，定义为非负正交ℝn+ 与ℝn 中标度为 m 的仿射子空间的交集。我们证明，一个简单且计算效率高的公式可以将 P 的体积逼近到 γm 的系数之内，其中 γ > 0 是一个绝对常量。该公式是目前已知的对运输多边形体积的最佳估计。
Let P be a bounded polyhedron defined as the intersection of the non-negative orthant ℝn+ and an affine subspace of codimension m in ℝn. We show that a simple and computationally efficient formula approximates the volume of P within a factor of γm, where γ > 0 is an absolute constant. The formula provides the best known estimate for the volume of transportation polytopes from a wide family.
期刊介绍：
The Israel Journal of Mathematics is an international journal publishing high-quality original research papers in a wide spectrum of pure and applied mathematics. The prestigious interdisciplinary editorial board reflects the diversity of subjects covered in this journal, including set theory, model theory, algebra, group theory, number theory, analysis, functional analysis, ergodic theory, algebraic topology, geometry, combinatorics, theoretical computer science, mathematical physics, and applied mathematics.
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