  A common way to quantify the ,,distance'' between measures is via their
discrepancy, also known as maximum mean discrepancy (MMD). Discrepancies are
related to Sinkhorn divergences $S_\varepsilon$ with appropriate cost functions
as $\varepsilon \to \infty$. In the opposite direction, if $\varepsilon \to 0$,
Sinkhorn divergences approach another important distance between measures,
namely the Wasserstein distance or more generally optimal transport
,,distance''. In this chapter, we investigate the limiting process for
arbitrary measures on compact sets and Lipschitz continuous cost functions. In
particular, we are interested in the behavior of the corresponding optimal
potentials $\hat \varphi_\varepsilon$, $\hat \psi_\varepsilon$ and $\hat
\varphi_K$ appearing in the dual formulation of the Sinkhorn divergences and
discrepancies, respectively. While part of the results are known, we provide
rigorous proofs for some relations which we have not found in this generality
in the literature. Finally, we demonstrate the limiting process by numerical
examples and show the behavior of the distances when used for the approximation
of measures by point measures in a process called dithering.
