The rival coffee shop problem

Abstract

In this paper, we will address a modification of the following optimization problem: given a positive integer $N$ and a compact Riemannian manifold X, the goal is to place a point $x_N\in X$ in such a way that the sequence $\{x_1,\dots,x_N\}\subset X$ is distributed as uniformly as possible, considering that $\{x_1,\dots,x_{N-1}\}\subset X$ already is. This can be thought as a way of placing coffee shops in a certain area one at a time in order to cover it optimally. So, following this modelization we will denote this problem as the coffee shop problem. This notion of optimal settlement is formalized in the context of optimal transport and Wasserstein distance. As a novel aspect, we introduce a new element to the problem: the presence of a rival brand, which competes against us by opening its own coffee shops. As our main tool, we use a variation of the Wasserstein distance, that allows us to work with finite signed measures and fits our problem. We present different results depending on how fast the rival is able to grow. With the Signed Wasserstein distance, we are able to obtain similar inequalities to the ones produced by the canonical Wasserstein one.