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 $\ell 2n$.
First, we show that there exist constants $\alpha ,ϵ>0$ such that for every positive integer $n$ there is a continuous odd function $\psi :Sm\to Sn$, with $m\ge \alpha n$, 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 $At$, which are related to the polarity transform $A$, is discussed. We prove that $At$ 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.