A Geometry-Based Approach for Solving the Transportation Problem with Euclidean Cost

In the semi-discrete version of Monge's problem one tries to find a transport map $T$ with minimum cost from an absolutely continuous measure $\mu$ on $\mathbb{R}^d$ to a discrete measure $\nu$ that is supported on a finite set in $\mathbb{R}^d$. The problem is considered for the case of the Euclidean cost function. Existence and uniqueness is shown by an explicit construction which yields a one-to-one mapping between the optimal $T$ and an additively weighted Voronoi partition of $\mathbb{R}^d$. From the proof an algorithm is derived to compute this partition.