{"id":147,"date":"2026-02-11T08:22:46","date_gmt":"2026-02-11T08:22:46","guid":{"rendered":"https:\/\/almoa.aau.at\/?page_id=63"},"modified":"2026-03-28T19:27:00","modified_gmt":"2026-03-28T18:27:00","slug":"dc-9-bounds-for-congruent-packings-of-convex-bodies-in-euclidean-space","status":"publish","type":"page","link":"https:\/\/almoa.aau.at\/?page_id=147","title":{"rendered":"DC 9 &#8211; Bounds for congruent packings of convex bodies in Euclidean space"},"content":{"rendered":"<div class=\"dc-project\">\r\n<p class=\"wp-block-paragraph\"><strong>Project Title:<\/strong> Bounds for congruent packings of convex bodies in Euclidean space<br \/><strong>Doctoral Candidate:<\/strong> N.N.<br \/><strong>Host Institution:<\/strong>\u00a0TU Delft<br \/><strong>Supervisors:<\/strong> <a href=\"https:\/\/diamhomes.ewi.tudelft.nl\/~fmario\/\">Fernando de Oliveira<\/a>, <a href=\"https:\/\/www.aau.at\/team\/wiegele-angelika\/\">Angelika Wiegele<\/a><\/p>\r\n\r\n\r\n\r\n<p class=\"wp-block-paragraph\"><strong>Objectives:<\/strong> Perhaps the most famous problem in discrete geometry is the sphere-packing problem: how densely can we pack nonoverlapping unit spheres in Euclidean space? In this project, we consider the more general case of congruential packings of convex bodies: how densely can we pack congruent copies (i.e., copies obtained by rotations and translations) of a given convex body in Euclidean space? The aim of this project is to develop theoretical and computational tools to obtain upper bounds for the density of congruential packings of convex bodies in Euclidean space. The more ambitious goal is to compute upper bounds for the density of packings of regular tetrahedra in 3-dimensional space. This will require an extension of the Cohn-Elkies bound for sphere packings, which was part of the recent breakthrough by the Fields medallist Maryna Viazovska, as well as the use of high-precision semidefinite programming (SDP) to solve the resulting 6-variable polynomial optimisation problem. The Supervisor has a lot of experience with such applications. A simpler problem to be considered along the way is packing pentagons in the plane. The optimal packing is known in this case, but a simpler numerical proof could perhaps be achieved with the use of semidefinite programming.<\/p>\r\n\r\n\r\n\r\n<p class=\"wp-block-paragraph\"><strong>Expected Results:<\/strong> A study of bounds for packings of polygons in the Euclidean plane and of how they behave. New bounds for packings of tetrahedra in 3-dimensional space. Computational tools to deal with such problems and the semidefinite programs coming from them.<\/p>\r\n\r\n\r\n\r\n<p class=\"wp-block-paragraph\"><strong>Planned secondment:<\/strong> 3 months at DLR (M. Epping) at the end of the 1st year to learn about applications of SDP in quantum computing; 3 months at UNI-KLU (A. Wiegele) during the 2nd year where the DC will work on implementing solution methods for the optimisation problems encountered during the research.<\/p>\r\n\r\n\r\n\r\n<p class=\"wp-block-paragraph\"><strong>Degree awarding institution:<\/strong> TU Delft<\/p>\r\n<p><strong>Note:<\/strong> Due to the 4-year PhD programme at TU Delft, the contract of DC 9 will be extended to 4 years.<\/p>\r\n<\/div>","protected":false},"excerpt":{"rendered":"<p>Project Title: Bounds for congruent packings of convex bodies in Euclidean spaceDoctoral Candidate: N.N.Host Institution:\u00a0TU DelftSupervisors: Fernando de Oliveira, Angelika Wiegele Objectives: Perhaps the most famous problem in discrete geometry is the sphere-packing problem: how densely can we pack nonoverlapping unit spheres in Euclidean space? In this project, we consider [&hellip;]<\/p>\n","protected":false},"author":3,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_crdt_document":"","footnotes":""},"class_list":["post-147","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/almoa.aau.at\/index.php?rest_route=\/wp\/v2\/pages\/147","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/almoa.aau.at\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/almoa.aau.at\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/almoa.aau.at\/index.php?rest_route=\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/almoa.aau.at\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=147"}],"version-history":[{"count":5,"href":"https:\/\/almoa.aau.at\/index.php?rest_route=\/wp\/v2\/pages\/147\/revisions"}],"predecessor-version":[{"id":878,"href":"https:\/\/almoa.aau.at\/index.php?rest_route=\/wp\/v2\/pages\/147\/revisions\/878"}],"wp:attachment":[{"href":"https:\/\/almoa.aau.at\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=147"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}