{"paper":{"title":"On the Inequalities of Projected Volumes and the Constructible Region","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DM","authors_text":"Liwei Zeng, Zihan Tan","submitted_at":"2014-10-31T08:11:05Z","abstract_excerpt":"We study the following geometry problem: given a $2^n-1$ dimensional vector $\\pi=\\{\\pi_S\\}_{S\\subseteq [n], S\\ne \\emptyset}$, is there an object $T\\subseteq\\mathbb{R}^n$ such that $\\log(\\mathsf{vol}(T_S))= \\pi_S$, for all $S\\subseteq [n]$, where $T_S$ is the projection of $T$ to the subspace spanned by the axes in $S$? If $\\pi$ does correspond to an object in $\\mathbb{R}^n$, we say that $\\pi$ is {\\em constructible}. We use $\\Psi_n$ to denote the constructible region, i.e., the set of all constructible vectors in $\\mathbb{R}^{2^n-1}$. In 1995, Bollob\\'{a}s and Thomason showed that $\\Psi_n$ is c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.8663","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}