{"paper":{"title":"Tight lower bounds on the number of faces of the Minkowski sum of convex polytopes via the Cayley trick","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.CG","authors_text":"Eleni Tzanaki, Menelaos I. Karavelas","submitted_at":"2011-12-07T12:00:03Z","abstract_excerpt":"Consider a set of $r$ convex $d$-polytopes $P_1,P_2,...,P_r$, where $d\\ge{}3$ and $r\\ge{}2$, and let $n_i$ be the number of vertices of $P_i$, $1\\le{}i\\le{}r$. It has been shown by Fukuda and Weibel that the number of $k$-faces of the Minkowski sum, $P_1+P_2+...+P_r$, is bounded from above by $\\Phi_{k+r}(n_1,n_2,...,n_r)$, where\n$\\Phi_{\\ell}(n_1,n_2,...,n_r)= \\sum_{\\substack{1\\le{}s_i\\le{}n_i\ns_1+...+s_r=\\ell}} \\prod_{i=1}^r\\binom{n_i}{s_i}$, $\\ell\\ge{}r$.\nFukuda and Weibel have also shown that the upper bound mentioned above is tight for $d\\ge{}4$, $2\\le{}r\\le{}\\lfloor\\frac{d}{2}\\rfloor$, and"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.1535","kind":"arxiv","version":1},"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"}