{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:56:00Z","timestamp":1787320560308,"version":"build-2736575974"},"reference-count":13,"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":[[2025,9,30]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>Let [Formula: see text] and let [Formula: see text] and [Formula: see text] be convex polytopes in [Formula: see text]. Denote by [Formula: see text] the set of midpoints of pairs [Formula: see text] with [Formula: see text], a vertex of [Formula: see text], and [Formula: see text], a vertex of [Formula: see text], such that [Formula: see text]. In the case [Formula: see text] and [Formula: see text], sharp lower bounds on [Formula: see text] in terms of the number of vertices of [Formula: see text] were given by Erd\u0151s, Furedi, and Fishburn in the early 1990s. Recently, Chestnut, Hildebrand, and Zenklusen gave nontrivial lower bounds on [Formula: see text] in terms of the number of vertices of [Formula: see text] for [Formula: see text]. On the one hand, the main statement of this paper generalizes the Erd\u0151s, Furedi, and Fishburn result for distinct convex polytopes. On the other hand, the main result of this paper improves the lower bounds obtained by Chestnut, Hildebrand, and Zenklusen for arbitrary [Formula: see text]. The main statement of this paper is a consequence of an additive combinatorics result. More specifically, it is a consequence of a Freiman cube lemma-type result which holds in arbitrary commutative groups, and it is interesting in its own right. Also, using other techniques, in this paper a slightly better bound of [Formula: see text] is reached when [Formula: see text]. Finally, our results have an application in a quantitative Helly-type theorem.<\/jats:p>","DOI":"10.1137\/24m1714496","type":"journal-article","created":{"date-parts":[[2025,9,10]],"date-time":"2025-09-10T07:21:46Z","timestamp":1757488906000},"page":"1806-1825","source":"Crossref","is-referenced-by-count":0,"title":["Midpoints of Vertex Pairs of Convex Polytopes"],"prefix":"10.1137","volume":"39","author":[{"given":"Mario","family":"Huicochea","sequence":"first","affiliation":[{"name":"CONAHCYT\/UAZ, Zacatecas."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Brien","family":"Navarro","sequence":"additional","affiliation":[{"name":"UAZ, Zacatecas."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2025,9,10]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1007\/s00493-015-3266-9"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1016\/j.aam.2017.04.003"},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.1090\/ulect\/077"},{"key":"ref4","first-page":"187","volume":"77","author":"Bell D. E.","year":"1976","journal-title":"Stud. Appl. Math."},{"key":"ref5","first-page":"77","volume":"258","author":"Bilu Y.","year":"1999","journal-title":"Structure Theory of Set Addition, Asterisque"},{"key":"ref6","doi-asserted-by":"publisher","DOI":"10.1137\/16M1100095"},{"key":"ref7","doi-asserted-by":"publisher","DOI":"10.1515\/advgeom-2017-0028"},{"key":"ref8","doi-asserted-by":"publisher","DOI":"10.1007\/BF01949705"},{"key":"ref9","doi-asserted-by":"publisher","DOI":"10.1137\/0404030"},{"key":"ref10","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4613-0039-7"},{"key":"ref11","volume-title":"Grad. Texts in Math. 165","author":"Nathanson M. B.","year":"1996"},{"key":"ref12","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.74.9.3637"},{"key":"ref13","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511755149"}],"container-title":["SIAM Journal on Discrete Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/24M1714496","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T12:57:45Z","timestamp":1787317065000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/24M1714496"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,9,10]]},"references-count":13,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2025,9,30]]}},"alternative-id":["10.1137\/24M1714496"],"URL":"https:\/\/doi.org\/10.1137\/24m1714496","relation":{},"ISSN":["0895-4801","1095-7146"],"issn-type":[{"value":"0895-4801","type":"print"},{"value":"1095-7146","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,9,10]]}}}