{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:45:11Z","timestamp":1787323511972,"version":"build-2736575974"},"reference-count":6,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Discrete Math."],"published-print":{"date-parts":[[2021,1]]},"abstract":"<jats:p>In this paper we study the following geometric problem: given $2^n-1$ real numbers $x_A$ indexed by the nonempty subsets $A\\subset \\{1,\\dots,n\\}$, is it possible to construct a body $T\\subset \\mathbb{R}^n$ such that $x_A=|T_A|$, where $|T_A|$ is the $|A|$-dimensional volume of the projection of $T$ onto the subspace spanned by the axes in $A$? As it is more convenient to take logarithms, we denote by $\\psi_n$ the set of all vectors $x$ for which there is a body $T$ such that $x_A=\\log |T_A|$ for all $A$. Bollob\u00e1s and Thomason showed that $\\psi_n$ is contained in the polyhedral cone defined by the class of \u201cuniform cover inequalities.\u201d Tan and Zeng conjectured that the convex hull $\\operatorname{conv}(\\psi_n)$ is equal to the cone given by the uniform cover inequalities. We prove that this conjecture is \u201cnearly\u201d right: the closed convex hull $\\overline{\\operatorname{conv}}(\\psi_n)$ is equal to the cone given by the uniform cover inequalities. However, perhaps surprisingly, we also show that $\\operatorname{conv}(\\psi_n)$ is not closed for $n\\ge 4$, thus disproving the conjecture.<\/jats:p>","DOI":"10.1137\/19m1296306","type":"journal-article","created":{"date-parts":[[2021,7,27]],"date-time":"2021-07-27T11:11:40Z","timestamp":1627384300000},"page":"1678-1687","source":"Crossref","is-referenced-by-count":1,"title":["Inequalities on Projected Volumes"],"prefix":"10.1137","volume":"35","author":[{"given":"Imre","family":"Leader","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0893-0347","authenticated-orcid":true,"given":"\u017darko","family":"Ran\u0111elovi\u0107","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Eero","family":"R\u00e4ty","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2021,7,27]]},"reference":[{"key":"atypb1","unstructured":"B. Bollob\u00e1s,\n                      Linear Analysis: An Introductory Course\n                      , Cambridge University Press, 1990."},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1112\/blms\/27.5.417"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1016\/0097-3165(86)90019-1"},{"key":"atypb4","doi-asserted-by":"crossref","unstructured":"D. Ellis, E. Friedgut, G. Kindler, and A. Yehudayoff,\n                      Geometric Stability via Information Theory\n                      , preprint,https:\/\/arxiv.org\/abs\/1510.00258, 2017.","DOI":"10.19086\/da.784"},{"key":"atypb5","unstructured":"D. G. Luenberger,\n                      Introduction to Linear and Non-Linear Programming\n                      , Addison-Wesley, 1984."},{"key":"atypb6","unstructured":"Z. Tan and L. Zeng,\n                      On the Inequalities of Projected Volumes and the Constructible Region\n                      , preprint,https:\/\/arxiv.org\/abs\/1410.8663, 2018."}],"container-title":["SIAM Journal on Discrete Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/19M1296306","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:06:44Z","timestamp":1787321204000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/19M1296306"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1]]},"references-count":6,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2021,1]]}},"alternative-id":["10.1137\/19M1296306"],"URL":"https:\/\/doi.org\/10.1137\/19m1296306","relation":{},"ISSN":["0895-4801","1095-7146"],"issn-type":[{"value":"0895-4801","type":"print"},{"value":"1095-7146","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,1]]}}}