{"id":{"repo_id":"cambridge","oai_identifier":"oai:www.repository.cam.ac.uk:1810/318559"},"canonical_url":"https://search.dev.ndltd.org/etd/cambridge/oai:www.repository.cam.ac.uk:1810/318559","repository":{"repo_id":"cambridge","name":"Cambridge University","base_url":"https://api.repository.cam.ac.uk/server/oai/request"},"display":{"title":"Topics in high-dimensional geometry and optimal transport","abstract":"The first two chapters of this thesis are devoted to a question of Vitali Milman about the existence of well-complemented almost Euclidean subspaces of spaces uniformly isomorphic to $\\ell_2^n$. First, we show that there exist constants $\\alpha,\\epsilon>0$ such that for every positive integer $n$ there is a continuous odd function $\\psi : S^m\\to S^n$, with $m\\geq \\alpha n$, such that the $\\epsilon$-expansion of the image of $\\psi$ does not contain a great circle. We also show how this result is connected to the aforementioned conjecture, more precisely that it allows to build a counterexample to a variation of the question. We then, in the second chapter, present an example of a normed space $X$ of arbitrarily high dimension that is strongly 2-Euclidean but contains no 2-dimensional subspace that is strongly $(1+\\epsilon)$-Euclidean and strongly $(1+\\epsilon)$-complemented, where $\\epsilon>0$ is an absolute constant. This is a counterexample to the ``strong'' Milman problem. The second part of this thesis involves topics related to optimal transport theory. The third chapter focuses on cost induced transforms. In particular, a family of order reversing isomorphisms $\\mathcal{A}_t$, which are related to the polarity transform $\\mathcal{A}$, is discussed. We prove that $\\mathcal{A}_t$ is the unique, up to linear terms, order reversing isomorphism on its image class. In the last chapter, we give a new proof of the Rockafellar-R\\\"uschendorf theorem about the existence of a potential for a given $c$-cyclically monotone set with a real-valued cost function. We then generalize the theorem to non-traditional cost functions, i.e. those which may also take the value $+\\infty$, and prove that a necessary and sufficient condition for the existence of a potential is that of $c$-path boundedness. Finally, we apply our theorem to show that for a continuous cost function and a compact, $c$-cyclically monotone set which is ``bounded away from infinity'' one gets a potential.","abstract_html":"The first two chapters of this thesis are devoted to a question of Vitali Milman about the existence of well-complemented almost Euclidean subspaces of spaces uniformly isomorphic to <span class=\"etd-inline-math\">\\ell<sub>2</sub><sup>n</sup></span>. First, we show that there exist constants <span class=\"etd-inline-math\">&alpha;,&epsilon;&gt;0</span> such that for every positive integer $n$ there is a continuous odd function <span class=\"etd-inline-math\">\\psi : S<sup>m</sup>\\to S<sup>n</sup></span>, with <span class=\"etd-inline-math\">m\\geq &alpha; n</span>, such that the <span class=\"etd-inline-math\">&epsilon;</span>-expansion of the image of $\\psi$ does not contain a great circle. We also show how this result is connected to the aforementioned conjecture, more precisely that it allows to build a counterexample to a variation of the question. We then, in the second chapter, present an example of a normed space $X$ of arbitrarily high dimension that is strongly 2-Euclidean but contains no 2-dimensional subspace that is strongly <span class=\"etd-inline-math\">(1+&epsilon;)</span>-Euclidean and strongly <span class=\"etd-inline-math\">(1+&epsilon;)</span>-complemented, where <span class=\"etd-inline-math\">&epsilon;&gt;0</span> is an absolute constant. This is a counterexample to the ``strong&#x27;&#x27; Milman problem. The second part of this thesis involves topics related to optimal transport theory. The third chapter focuses on cost induced transforms. In particular, a family of order reversing isomorphisms <span class=\"etd-inline-math\">\\mathcal{A}<sub>t</sub></span>, which are related to the polarity transform $\\mathcal{A}$, is discussed. We prove that <span class=\"etd-inline-math\">\\mathcal{A}<sub>t</sub></span> is the unique, up to linear terms, order reversing isomorphism on its image class. In the last chapter, we give a new proof of the Rockafellar-R\\&quot;uschendorf theorem about the existence of a potential for a given $c$-cyclically monotone set with a real-valued cost function. We then generalize the theorem to non-traditional cost functions, i.e. those which may also take the value $+\\infty$, and prove that a necessary and sufficient condition for the existence of a potential is that of $c$-path boundedness. Finally, we apply our theorem to show that for a continuous cost function and a compact, $c$-cyclically monotone set which is ``bounded away from infinity&#x27;&#x27; one gets a potential.","abstract_has_math":true,"creators":["Wyczesany, Katarzyna"],"institution":"University of Cambridge","degree_name":"Doctor of Philosophy (PhD)","degree_level":"Doctoral","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Gowers, William Timothy"],"committee_chairs":[],"committee_members":[],"year":2020,"date_issued":"2020-10-28","date_published":"2020-10-28","updated_at":"2026-07-22T22:24:28Z","subjects":["Geometric Functional Analysis","Optimal Transport","Normed Space","Order-reversing Isomorphism","Potential"],"languages":["eng"],"rights":[],"rights_urls":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/9e27bb82-bdbd-40ef-bd5d-c2add8d00282/download","https://www.rioxx.net/licenses/all-rights-reserved/"],"identifier_entries":[{"key":"dc:creator.authoridentifier","label":"Author Identifier","values":["0000000215307916"],"render_values":[{"text":"0000-0002-1530-7916","href":"https://orcid.org/0000-0002-1530-7916","code":true}]}]},"links":{"outbound_url":"https://doi.org/10.17863/CAM.65672","outbound_label":"DOI","outbound_source":"dc:identifier.doi"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Gowers, William Timothy"]},{"key":"dc:contributor.sponsor","label":"Sponsor","values":["UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/L016516/1 for the Cambridge Centre for Analysis (CCA)."]},{"key":"dc:creator","label":"Author","values":["Wyczesany, Katarzyna"]},{"key":"dc:creator.authoridentifier","label":"Author Identifier","values":["0000000215307916"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.issued","label":"Date","values":["2020-10-28"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["University of Cambridge"]},{"key":"dc:relation.isreferencedby.uri","label":"Dc Relation Isreferencedby URI","values":["https://www.repository.cam.ac.uk/handle/1810/318559"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["Doctoral"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["Doctor of Philosophy (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Geometric Functional Analysis","Optimal Transport","Normed Space","Order-reversing Isomorphism","Potential"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/9e27bb82-bdbd-40ef-bd5d-c2add8d00282/download","https://www.rioxx.net/licenses/all-rights-reserved/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.doi","label":"DOI","values":["10.17863/CAM.65672"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/9db0e651-cfc6-46e5-9a35-f5e04c37c9e4/download"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["The first two chapters of this thesis are devoted to a question of Vitali Milman about the existence of well-complemented almost Euclidean subspaces of spaces uniformly isomorphic to $\\ell_2^n$. 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In particular, a family of order reversing isomorphisms $\\mathcal{A}_t$, which are related to the polarity transform $\\mathcal{A}$, is discussed. We prove that $\\mathcal{A}_t$ is the unique, up to linear terms, order reversing isomorphism on its image class. In the last chapter, we give a new proof of the Rockafellar-R\\\"uschendorf theorem about the existence of a potential for a given $c$-cyclically monotone set with a real-valued cost function. We then generalize the theorem to non-traditional cost functions, i.e. those which may also take the value $+\\infty$, and prove that a necessary and sufficient condition for the existence of a potential is that of $c$-path boundedness. Finally, we apply our theorem to show that for a continuous cost function and a compact, $c$-cyclically monotone set which is ``bounded away from infinity'' one gets a potential."]},{"key":"dc:format.checksum.md5","label":"Dc Format Checksum Md5","values":["10d860c5a619e4f8852c76f58f066495","353adac0d1ebdfd65ab16480263c3c87"]},{"key":"dc:title","label":"Title","values":["Topics in high-dimensional geometry and optimal transport"]}]}],"canonical_facts":{"dc:contributor.advisor":["Gowers, William Timothy"],"dc:contributor.sponsor":["UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/L016516/1 for the Cambridge Centre for Analysis (CCA)."],"dc:creator":["Wyczesany, Katarzyna"],"dc:creator.authoridentifier":["0000000215307916"],"dc:date.issued":["2020-10-28"],"dc:description.abstract":["The first two chapters of this thesis are devoted to a question of Vitali Milman about the existence of well-complemented almost Euclidean subspaces of spaces uniformly isomorphic to $\\ell_2^n$. First, we show that there exist constants $\\alpha,\\epsilon>0$ such that for every positive integer $n$ there is a continuous odd function $\\psi : S^m\\to S^n$, with $m\\geq \\alpha n$, such that the $\\epsilon$-expansion of the image of $\\psi$ does not contain a great circle. We also show how this result is connected to the aforementioned conjecture, more precisely that it allows to build a counterexample to a variation of the question. We then, in the second chapter, present an example of a normed space $X$ of arbitrarily high dimension that is strongly 2-Euclidean but contains no 2-dimensional subspace that is strongly $(1+\\epsilon)$-Euclidean and strongly $(1+\\epsilon)$-complemented, where $\\epsilon>0$ is an absolute constant. This is a counterexample to the ``strong'' Milman problem. The second part of this thesis involves topics related to optimal transport theory. The third chapter focuses on cost induced transforms. In particular, a family of order reversing isomorphisms $\\mathcal{A}_t$, which are related to the polarity transform $\\mathcal{A}$, is discussed. We prove that $\\mathcal{A}_t$ is the unique, up to linear terms, order reversing isomorphism on its image class. In the last chapter, we give a new proof of the Rockafellar-R\\\"uschendorf theorem about the existence of a potential for a given $c$-cyclically monotone set with a real-valued cost function. We then generalize the theorem to non-traditional cost functions, i.e. those which may also take the value $+\\infty$, and prove that a necessary and sufficient condition for the existence of a potential is that of $c$-path boundedness. Finally, we apply our theorem to show that for a continuous cost function and a compact, $c$-cyclically monotone set which is ``bounded away from infinity'' one gets a potential."],"dc:format.checksum.md5":["10d860c5a619e4f8852c76f58f066495","353adac0d1ebdfd65ab16480263c3c87"],"dc:identifier.doi":["10.17863/CAM.65672"],"dc:identifier.uri":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/9db0e651-cfc6-46e5-9a35-f5e04c37c9e4/download"],"dc:language":["eng"],"dc:publisher.institution":["University of Cambridge"],"dc:relation.isreferencedby.uri":["https://www.repository.cam.ac.uk/handle/1810/318559"],"dc:rights":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/9e27bb82-bdbd-40ef-bd5d-c2add8d00282/download","https://www.rioxx.net/licenses/all-rights-reserved/"],"dc:subject":["Geometric Functional Analysis","Optimal Transport","Normed Space","Order-reversing Isomorphism","Potential"],"dc:title":["Topics in high-dimensional geometry and optimal transport"],"dc:type":["Thesis"],"dc:type.qualificationlevel":["Doctoral"],"dc:type.qualificationname":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-22T22:24:28Z"}