## Topics in high-dimensional geometry and optimal transport

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##### Authors

##### Advisors

Gowers, William Timothy

##### Date

2020-10-28##### Awarding Institution

University of Cambridge

##### Author Affiliation

Dept of Pure Mathematics and Mathematical Statistics

##### Qualification

Doctor of Philosophy (PhD)

##### Type

Thesis

##### Metadata

Show full item record##### Citation

Wyczesany, K. (2020). Topics in high-dimensional geometry and optimal transport (Doctoral thesis). https://doi.org/10.17863/CAM.65672

##### 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.

##### Keywords

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

##### Sponsorship

UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/L016516/1 for the Cambridge Centre for Analysis (CCA).

##### Funder references

EPSRC (1804242)

##### Identifiers

This record's DOI: https://doi.org/10.17863/CAM.65672

##### Rights

All rights reserved

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