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Topics in high-dimensional geometry and optimal transport 10.17863/CAM.65672 Wyczesany, Katarzyna 0000000215307916 Gowers, William Timothy 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. University of Cambridge 2020-10-28 Thesis Doctoral Doctor of Philosophy (PhD) eng UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/L016516/1 for the Cambridge Centre for Analysis (CCA). https://www.repository.cam.ac.uk/handle/1810/318559 https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/9db0e651-cfc6-46e5-9a35-f5e04c37c9e4/download 10d860c5a619e4f8852c76f58f066495 https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/9e27bb82-bdbd-40ef-bd5d-c2add8d00282/download 353adac0d1ebdfd65ab16480263c3c87 https://www.rioxx.net/licenses/all-rights-reserved/ Geometric Functional Analysis Optimal Transport Normed Space Order-reversing Isomorphism Potential