Topics in high-dimensional geometry and optimal transport
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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
First, we show that there exist constants
We then, in the second chapter, present an example of a normed space
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
In the last chapter, we give a new proof of the Rockafellar-R"uschendorf theorem about the existence of a potential for a given