{"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":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s11856-024-2615-z","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
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.
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