{"id":{"repo_id":"ohiolink","oai_identifier":"oai:etd.ohiolink.edu:case1365179413"},"canonical_url":"https://search.dev.ndltd.org/etd/ohiolink/oai:etd.ohiolink.edu:case1365179413","repository":{"repo_id":"ohiolink","name":"OhioLINK","base_url":"https://etd.ohiolink.edu/acprod/odb_etd/ws/oai/oai"},"display":{"title":"Convex Geometric Connections to Information Theory","abstract":"Convex geometry is a field of mathematics that has experienced rapid growth in recent years and has proven to be an extremely useful perspective in areas of research. Problems in many different fields can be interpreted geometrically which often leads to powerful and surprising results. This thesis establishes connections between convex geometry and both classical and quantum information theory. We introduce the mean width bodies and illustrate the geometric interpretation they provide for the relative entropy of cone measures of a convex body and its polar. We define relative entropy for convex bodies and its relation to affine isoperimetric inequalities is considered. Other connections are made by considering quantum information theory. The fundamental objects in quantum information theory are quantum states. The set of states is convex as are some of its important subsets. Therefore, convex geometry provides a natural approach to explore quantum states. Fairly sharp estimates are obtained regarding the geometry of quantum states using basic notions in convex geometry. In particular, the distance between the set of states with positive partial transpose and the set of separable states is explored. Finally, the optimal constants for the spherical isoperimetric inequality are provided and generalizations of a concentration inequality are suggested.","abstract_html":"Convex geometry is a field of mathematics that has experienced rapid growth in recent years and has proven to be an extremely useful perspective in areas of research. Problems in many different fields can be interpreted geometrically which often leads to powerful and surprising results. This thesis establishes connections between convex geometry and both classical and quantum information theory. We introduce the mean width bodies and illustrate the geometric interpretation they provide for the relative entropy of cone measures of a convex body and its polar. We define relative entropy for convex bodies and its relation to affine isoperimetric inequalities is considered. Other connections are made by considering quantum information theory. The fundamental objects in quantum information theory are quantum states. The set of states is convex as are some of its important subsets. Therefore, convex geometry provides a natural approach to explore quantum states. Fairly sharp estimates are obtained regarding the geometry of quantum states using basic notions in convex geometry. In particular, the distance between the set of states with positive partial transpose and the set of separable states is explored. Finally, the optimal constants for the spherical isoperimetric inequality are provided and generalizations of a concentration inequality are suggested.","abstract_has_math":false,"creators":["Jenkinson, Justin"],"institution":"Case Western Reserve University School of Graduate Studies","degree_name":"Doctor of Philosophy","degree_level":"doctoral","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Szarek, Stanislaw","Werner, Elisabeth"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2013,"date_issued":"2013-08-16","date_published":"2013-08-16","updated_at":"2026-07-24T03:37:31Z","subjects":["Mathematics"],"languages":["English"],"rights":["unrestricted","This thesis or dissertation is protected by copyright: all rights reserved. It may not be copied or redistributed beyond the terms of applicable copyright laws."],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://rave.ohiolink.edu/etdc/view?acc_num=case1365179413","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Szarek, Stanislaw","Werner, Elisabeth"]},{"key":"dc:creator","label":"Author","values":["Jenkinson, Justin"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2013-08-16"]},{"key":"dc:publisher","label":"Institution","values":["Case Western Reserve University School of Graduate Studies / OhioLINK"]},{"key":"dc:type","label":"Dc Type","values":["Electronic Thesis or Dissertation"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["doctoral"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Case Western Reserve University School of Graduate Studies"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["English"]},{"key":"dc:rights","label":"Dc Rights","values":["unrestricted","This thesis or dissertation is protected by copyright: all rights reserved. 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We define relative entropy for convex bodies and its relation to affine isoperimetric inequalities is considered. Other connections are made by considering quantum information theory. The fundamental objects in quantum information theory are quantum states. The set of states is convex as are some of its important subsets. Therefore, convex geometry provides a natural approach to explore quantum states. Fairly sharp estimates are obtained regarding the geometry of quantum states using basic notions in convex geometry. In particular, the distance between the set of states with positive partial transpose and the set of separable states is explored. Finally, the optimal constants for the spherical isoperimetric inequality are provided and generalizations of a concentration inequality are suggested."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf","2.2 MB"]},{"key":"dc:title","label":"Title","values":["Convex Geometric Connections to Information Theory"]}]}],"canonical_facts":{"dc:contributor":["Szarek, Stanislaw","Werner, Elisabeth"],"dc:creator":["Jenkinson, Justin"],"dc:date":["2013-08-16"],"dc:description":["Convex geometry is a field of mathematics that has experienced rapid growth in recent years and has proven to be an extremely useful perspective in areas of research. Problems in many different fields can be interpreted geometrically which often leads to powerful and surprising results. This thesis establishes connections between convex geometry and both classical and quantum information theory. We introduce the mean width bodies and illustrate the geometric interpretation they provide for the relative entropy of cone measures of a convex body and its polar. We define relative entropy for convex bodies and its relation to affine isoperimetric inequalities is considered. Other connections are made by considering quantum information theory. The fundamental objects in quantum information theory are quantum states. The set of states is convex as are some of its important subsets. Therefore, convex geometry provides a natural approach to explore quantum states. Fairly sharp estimates are obtained regarding the geometry of quantum states using basic notions in convex geometry. In particular, the distance between the set of states with positive partial transpose and the set of separable states is explored. Finally, the optimal constants for the spherical isoperimetric inequality are provided and generalizations of a concentration inequality are suggested."],"dc:format":["application/pdf","2.2 MB"],"dc:identifier":["http://rave.ohiolink.edu/etdc/view?acc_num=case1365179413"],"dc:language":["English"],"dc:publisher":["Case Western Reserve University School of Graduate Studies / OhioLINK"],"dc:rights":["unrestricted","This thesis or dissertation is protected by copyright: all rights reserved. It may not be copied or redistributed beyond the terms of applicable copyright laws."],"dc:subject":["Mathematics"],"dc:title":["Convex Geometric Connections to Information Theory"],"dc:type":["Electronic Thesis or Dissertation"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["doctoral"],"thesis:degree_name":["Doctor of Philosophy"],"thesis:institution_name":["Case Western Reserve University School of Graduate Studies"]},"updated_at":"2026-07-24T03:37:31Z"}