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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1112.1535","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CG","submitted_at":"2011-12-07T12:00:03Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"04f64799eae9010aef1143b1cf57be46904143cb34ba3d40cb8c111e81616e00","abstract_canon_sha256":"bf3c781a7deb9dd75217ed95241bb497edcb2d2d68265692e425d985c20011aa"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:06:54.395099Z","signature_b64":"JzVME9mXj0UJa5nA9mP2hoIwM+tIryF4YSwCJBXTaq9NUEdicP7klAomCjWReIGQBxiFuzhTZ09gGfmAHe/mCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5e9f6e8bde7a95490f17bfa5e4aaa37ee438d70d8cbe9c2a1bc031d7854815b1","last_reissued_at":"2026-05-18T04:06:54.394623Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:06:54.394623Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $P_i$, $1\\le{}i\\le{}r$. 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