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The rival coffee shop problem

Javier Casado, Manuel Cuerno

Abstract

We show some estimations for the Coffee Shop Problem with a modification respect the original statement: there is a rival competing against us. We present different results based on how fast the rival is able to grow. As main tool, we use a variation of the Wasserstein distance (the Signed Wasserstein Distance presented by Piccoli, Rossi and Tournus in "A Wasserstein norm for signed measures, with application to non local transport equation with source term"), that allow us to work with finite signed measures and fits our problem. We are able to obtain similar inequalities with this distance to the ones produced by the canonical Wasserstein distance.

The rival coffee shop problem

Abstract

We show some estimations for the Coffee Shop Problem with a modification respect the original statement: there is a rival competing against us. We present different results based on how fast the rival is able to grow. As main tool, we use a variation of the Wasserstein distance (the Signed Wasserstein Distance presented by Piccoli, Rossi and Tournus in "A Wasserstein norm for signed measures, with application to non local transport equation with source term"), that allow us to work with finite signed measures and fits our problem. We are able to obtain similar inequalities with this distance to the ones produced by the canonical Wasserstein distance.
Paper Structure (9 sections, 14 theorems, 60 equations, 3 figures)

This paper contains 9 sections, 14 theorems, 60 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Generalizing the Wasserstein distance
  3. Fixed competition
  4. Non--fixed competition
  5. Forbidden areas
  6. Rival growth in terms of ours
  7. Case $f(N)\geq f(N-1)+2$
  8. Case $f(N)= N+K$
  9. Discussion about the constants in the difference of Dirac deltas

Key Result

Theorem 1.1

Let $X$ be a smooth, compact $d$--dimensional manifold without boundary. Then, for any set of $N$ points $\{x_1,...,x_N\}\subset X$, the following inequality holds where $G:X\times X\to\mathbb{R}\cup\{\infty\}$ denote the Green's function of the Laplacian normalized to have average value $0$ over the manifold and $d\geq3$. If the manifold is two--dimensional, a slightly different inequality also h

Figures (3)

  • Figure 1:
  • Figure 2: On the left: one Coffee shop with the setup of Figure \ref{['figure1']}. On the right: three Coffee shops with the setup of Figure \ref{['figure1']}
  • Figure 3: On the left: the settlement of the first Coffee shop. On the right: a settlement for four Coffee shops that seems to fit better the coffee shop problem than Figure \ref{['figure1']}

Theorems & Definitions (30)

  • Theorem 1.1: Steinerberger, steinerbergermanifold
  • Remark
  • Theorem \ref{teoremauno}
  • Theorem \ref{thm33}
  • Remark
  • Theorem \ref{proposicioncon2N}
  • Theorem \ref{thm45}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3: Signed Generalized Wasserstein distance, mainini,piccoli
  • ...and 20 more