{"id":104,"date":"2010-03-30T06:00:48","date_gmt":"2010-03-30T10:00:48","guid":{"rendered":"http:\/\/blogs.williams.edu\/Morgan\/?p=104"},"modified":"2010-03-30T06:00:48","modified_gmt":"2010-03-30T10:00:48","slug":"isoperimetric-inequality-in-complement-of-mean-convex-set-fails-at-banff","status":"publish","type":"post","link":"https:\/\/sites.williams.edu\/Morgan\/2010\/03\/30\/isoperimetric-inequality-in-complement-of-mean-convex-set-fails-at-banff\/","title":{"rendered":"Isoperimetric Inequality in Complement of Mean Convex Set Fails at Banff"},"content":{"rendered":"<div style=\"background-color: #ffffff;font: normal normal normal 13px\/19px Georgia, 'Times New Roman', 'Bitstream Charter', Times, serif;padding: 0.6em;margin: 0px\">\n<p>On March 29 at\u00a0<a href=\"http:\/\/www.birs.ca\/birspages.php?task=displayevent&amp;event_id=10w5114\">Banff<\/a>, Mohammad\u00a0<a href=\"http:\/\/www.math.gatech.edu\/users\/ghomi\">Ghomi<\/a> talked on his proof [CGR] with Choe and Ritor\u00e9 of the isoperimetric inequality in the complement of a convex body<em> K<\/em> in\u00a0R<sup>n<\/sup>: the area of a hypersurface enclosing volume\u00a0<em>V<\/em> outside the convex body is at least the area of a hemisphere of volume\u00a0<em>V<\/em>. I asked whether it suffices to assume<em> K<\/em> mean convex (nonnegative mean curvature). The answer is no. <!--more-->The first counterexample, provided by Cantarella and Ghomi, was\u00a0<em>K<\/em> a catenoid and the hypersurface a portion of a cylinder of large radius. What about for small topological balls? The answer is still no. Otherwise a small ball split in half by a minimal surface would have to have equal perimeter on each side, each half would be isoperimetric, the sphere would meet the minimal surface normally, and the minimal surface would have to be a cone.<\/p>\n<p><a href=\"http:\/\/www.ams.org\/mathscinet\/search\/publdoc.html?arg3=&amp;co4=AND&amp;co5=AND&amp;co6=AND&amp;co7=AND&amp;dr=all&amp;pg4=AUCN&amp;pg5=AUCN&amp;pg6=AUCN&amp;pg7=ALLF&amp;pg8=ET&amp;review_format=html&amp;s4=ghomi&amp;s5=ritore&amp;s6=choe&amp;s7=&amp;s8=All&amp;vfpref=html&amp;yearRangeFirst=&amp;yearRangeSecond=&amp;yrop=eq&amp;r=1&amp;mx-pid=2329803\">[CGR]<\/a> Choe, Jaigyoung(KR-KIAS-SM); Ghomi, Mohammad(1-GAIT); Ritor\u00e9, Manuel(E-GRANS-GT)<br \/>\nThe relative isoperimetric inequality outside convex domains in R<sup>n<\/sup>.\u00a0Calc. Var. Partial Differential Equations 29 (2007), no. 4, 421-429.<\/p>\n<p><strong>P.S. May, 2010.<\/strong> <em>Pacific Journal<\/em> (Vol. 244, No. 2, 2010) has an article by Mouhamed Moustapha Fall on &#8220;Area-minimizing regions with small volume in Riemannian manifolds with boundary.&#8221;\u00a0The main result is that inside a smooth, bounded region with boundary in R<sup>n<\/sup> or in any Riemannian manifold, small isoperimetric regions are approximate half-balls near boundary points of maximum mean curvature H. (Should also apply to unbounded region such as complement of smooth body.) Moreover, if another domain has H&#8217; &lt; H, then for small volume the isoperimetric profiles satisfy I&#8217; &gt; I. In particular, outside a smooth strictly mean convex 3D region, for small volumes, area is greater than that of the Euclidean half-ball of the same volume. This does not contradict the nearly opposite conclusion we came to above\u00a0because of the strict inequality in the hypothesis H&#8217; &lt; H and the restriction to small volumes.<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>On March 29 at\u00a0Banff, Mohammad\u00a0Ghomi talked on his proof [CGR] with Choe and Ritor\u00e9 of the isoperimetric inequality in the complement of a convex body K in\u00a0Rn: the area of a hypersurface enclosing volume\u00a0V outside the convex body is at least the area of a hemisphere of volume\u00a0V. I asked whether it suffices to assume [&hellip;]<\/p>\n","protected":false},"author":269,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[14043],"tags":[],"class_list":["post-104","post","type-post","status-publish","format-standard","hentry","category-math"],"acf":[],"_links":{"self":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts\/104","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/users\/269"}],"replies":[{"embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/comments?post=104"}],"version-history":[{"count":0,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts\/104\/revisions"}],"wp:attachment":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/media?parent=104"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/categories?post=104"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/tags?post=104"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}