{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article7855","name":"A Geometric Approach for the Upper Bound Theorem for Minkowski Sums of Convex Polytopes","abstract":"We derive tight expressions for the maximum number of k-faces, k=0,...,d-1, of the Minkowski sum, P_1+...+P_r, of r convex d-polytopes P_1,...,P_r in R^d, where d >= 2 and r < d, as a (recursively defined) function on the number of vertices of the polytopes. Our results coincide with those recently proved by Adiprasito and Sanyal [1]. In contrast to Adiprasito and Sanyal's approach, which uses tools from Combinatorial Commutative Algebra, our approach is purely geometric and uses basic notions such as f- and h-vector calculus, stellar subdivisions and shellings, and generalizes the methodology used in [10] and [9] for proving upper bounds on the f-vector of the Minkowski sum of two and three convex polytopes, respectively. The key idea behind our approach is to express the Minkowski sum P_1+...+P_r as a section of the Cayley polytope C of the summands; bounding the k-faces of P_1+...+P_r reduces to bounding the subset of the (k+r-1)-faces of C that contain vertices from each of the r polytopes. We end our paper with a sketch of an explicit construction that establishes the tightness of the upper bounds.","keywords":["Convex polytopes","Minkowski sum","upper bound"],"author":[{"@type":"Person","name":"Karavelas, Menelaos I.","givenName":"Menelaos I.","familyName":"Karavelas"},{"@type":"Person","name":"Tzanaki, Eleni","givenName":"Eleni","familyName":"Tzanaki"}],"position":10,"pageStart":81,"pageEnd":95,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Karavelas, Menelaos I.","givenName":"Menelaos I.","familyName":"Karavelas"},{"@type":"Person","name":"Tzanaki, Eleni","givenName":"Eleni","familyName":"Tzanaki"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.81","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":["http:\/\/arxiv.org\/abs\/1405.7368v3","http:\/\/arxiv.org\/abs\/1502.02265v2"],"isPartOf":{"@type":"PublicationVolume","@id":"#volume6237","volumeNumber":34,"name":"31st International Symposium on Computational Geometry (SoCG 2015)","dateCreated":"2015-06-12","datePublished":"2015-06-12","editor":[{"@type":"Person","name":"Arge, Lars","givenName":"Lars","familyName":"Arge"},{"@type":"Person","name":"Pach, J\u00e1nos","givenName":"J\u00e1nos","familyName":"Pach"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article7855","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6237"}}}