{"id":190629,"date":"2003-08-19T00:00:12","date_gmt":"2003-08-19T07:00:12","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/a-topological-colorful-helly-theorem\/"},"modified":"2016-09-09T15:29:30","modified_gmt":"2016-09-09T22:29:30","slug":"a-topological-colorful-helly-theorem","status":"publish","type":"msr-video","link":"https:\/\/www.microsoft.com\/en-us\/research\/video\/a-topological-colorful-helly-theorem\/","title":{"rendered":"A topological colorful Helly theorem"},"content":{"rendered":"<div class=\"asset-content\">\n<p>Let F be a finite family of convex sets in n-dimensional Euclidean space. Helly&#8217;s theorem asserts that if every n+1 members of the family has a point in common then there is a point in common to all members of the family.<\/p>\n<p>Lovasz proved the following extension of Helly theorem: Colorful Helly Theorem: Consider n+1 families of convex sets in R<sup>n<\/sup> and suppose that for every choice of n+1 sets, one from each family there is a point in common to all these sets. Then one of the families is intersecting.<\/p>\n<p>This innocent looking extension is quite deeper than Helly&#8217;s original theorem and the associated Caratheodory-type theorem of Barany has many applications in discrete geometry.<\/p>\n<p>Helly himself realized that in his theorem convex sets can be replaced by topological cells if you impose the requirement that all non-empty intersections of these sets are again topological cells. (Since the intersection of convex sets is also convex this requirement is automatically satisfied in the original geometric version.) Helly&#8217;s topological version of his theorem follows from the later &#8220;nerve theorems&#8221; of Leray and others.<\/p>\n<p>In the lecture I will present a topological version for Lovasz&#8217; colorful Helly theorem and discuss related issues.<\/p>\n<p>Gil Kalai, joint work with Roy Meshulam<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Let F be a finite family of convex sets in n-dimensional Euclidean space. Helly&#8217;s theorem asserts that if every n+1 members of the family has a point in common then there is a point in common to all members of the family. Lovasz proved the following extension of Helly theorem: Colorful Helly Theorem: Consider n+1 [&hellip;]<\/p>\n","protected":false},"featured_media":198231,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr_hide_image_in_river":0,"footnotes":""},"research-area":[],"msr-video-type":[],"msr-locale":[268875],"msr-post-option":[],"msr-session-type":[],"msr-impact-theme":[],"msr-pillar":[],"msr-episode":[],"msr-research-theme":[],"class_list":["post-190629","msr-video","type-msr-video","status-publish","has-post-thumbnail","hentry","msr-locale-en_us"],"msr_download_urls":"","msr_external_url":"https:\/\/youtu.be\/O5y9riv7qQU","msr_secondary_video_url":"","msr_video_file":"","_links":{"self":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/190629","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video"}],"about":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/types\/msr-video"}],"version-history":[{"count":0,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/190629\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media\/198231"}],"wp:attachment":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media?parent=190629"}],"wp:term":[{"taxonomy":"msr-research-area","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/research-area?post=190629"},{"taxonomy":"msr-video-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video-type?post=190629"},{"taxonomy":"msr-locale","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-locale?post=190629"},{"taxonomy":"msr-post-option","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-post-option?post=190629"},{"taxonomy":"msr-session-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-session-type?post=190629"},{"taxonomy":"msr-impact-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-impact-theme?post=190629"},{"taxonomy":"msr-pillar","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-pillar?post=190629"},{"taxonomy":"msr-episode","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-episode?post=190629"},{"taxonomy":"msr-research-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-theme?post=190629"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}