{"id":20967,"date":"2026-09-11T17:10:56","date_gmt":"2026-09-11T15:10:56","guid":{"rendered":"https:\/\/scm.iec.cat\/?post_type=conference&#038;p=20967"},"modified":"2026-09-12T16:06:12","modified_gmt":"2026-09-12T14:06:12","slug":"2026-tfm-jaume-capdevila","status":"publish","type":"conference","link":"https:\/\/scm.iec.cat\/eng\/conference\/2026-tfm-jaume-capdevila\/","title":{"rendered":"2026-TFM-Jaume Capdevila"},"content":{"rendered":"<h5>Abstract<\/h5>\n<div>One of the main concepts of geometric measure theory is that of a rectifiable set, and one of its main objectives is to characterize them by means of geometric or analytic properties. One possible characterization is in terms of densities. Given integers 0 &lt; d &lt; n and a set E of R<sup>n<\/sup>, the lower d-density of E at a point x of R<sup>n<\/sup> is defined as the lower limit as r tends to 0 of the quotient between the d-dimensional Hausdorff measure of E \u2229 B(x,r), which we denote H(E \u2229 B(x,r)), and (2r)^d. In 1938, Besicovitch proved that given a set E of the plane R<sup>2<\/sup>, if the lower 1-density of E is strictly greater than 3\/4 at almost all of its points, then E is 1-rectifiable. He also gave an example of a purely 1-unrectifiable set of the plane R<sup>2<\/sup>, with lower density equal to 1\/2 at almost all of its points.<\/div>\n<div><\/div>\n<div><\/div>\n<div>In this presentation, we present a generalization to arbitrary dimensions of the example originally introduced by Besicovitch. In this way, we obtain a purely d-unrectifiable set in R<sup>d+1<\/sup> with lower d-density equal to 1\/2 at almost all of its points. This establishes the lower bound 1\/2 for the minimum value S such that if the lower d-density of a set E is strictly greater than S at all of its points, then E is d-rectifiable.<\/div>\n<div><\/div>\n<div><\/div>","protected":false},"featured_media":0,"parent":0,"template":"","categoria_sessions_conferences":[187],"class_list":["post-20967","conference","type-conference","status-publish","hentry","categoria_sessions_conferences-2a-jornada-tfm-2024"],"acf":[],"_links":{"self":[{"href":"https:\/\/scm.iec.cat\/eng\/wp-json\/wp\/v2\/conference\/20967","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/scm.iec.cat\/eng\/wp-json\/wp\/v2\/conference"}],"about":[{"href":"https:\/\/scm.iec.cat\/eng\/wp-json\/wp\/v2\/types\/conference"}],"wp:attachment":[{"href":"https:\/\/scm.iec.cat\/eng\/wp-json\/wp\/v2\/media?parent=20967"}],"wp:term":[{"taxonomy":"categoria_sessions_conferences","embeddable":true,"href":"https:\/\/scm.iec.cat\/eng\/wp-json\/wp\/v2\/categoria_sessions_conferences?post=20967"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}