Topics in high-dimensional geometry and optimal transport

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 \ell₂ⁿ. First, we show that there exist constants α,ε>0 such that for every positive integer $n$ there is a continuous odd function \psi : S^m\to Sⁿ, with m\geq α n, such that the ε-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+ε)-Euclidean and strongly (1+ε)-complemented, where ε>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}<sub>t</sub>, which are related to the polarity transform $\mathcal{A}$, is discussed. We prove that \mathcal{A}<sub>t</sub> 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.

Name dc:type.qualificationname

Doctor of Philosophy (PhD)

Level dc:type.qualificationlevel

Doctoral

Grantor dc:publisher.institution

University of Cambridge

Year dc:date.issued

2020

Author dc:creator

• Wyczesany, Katarzyna

Advisor dc:contributor.advisor

• Gowers, William Timothy

• Geometric Functional Analysis
• Optimal Transport
• Normed Space
• Order-reversing Isomorphism
• Potential

Licence

• https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/9e27bb82-bdbd-40ef-bd5d-c2add8d00282/download
• https://www.rioxx.net/licenses/all-rights-reserved/

Language dc:language

eng

Author Identifier

0000-0002-1530-7916

OAI identifier oai:identifier

oai:www.repository.cam.ac.uk:1810/318559