A uniform betweenness property in metric spaces and its role in the quantitative analysis of the “Lion-Man” game

Ulrich Kohlenbach, Genaro López-Acedo and Adriana Nicolae

Vol. 310 (2021), No. 1, 181–212

DOI: 10.2140/pjm.2021.310.181

Abstract

We analyze, based on an interplay between ideas and techniques from logic and geometric analysis, a pursuit-evasion game. More precisely, we focus on a uniform betweenness property and use it in the study of a discrete lion and man game with an $𝜀$-capture criterion. In particular, we prove that in uniformly convex bounded domains the lion always wins and, using ideas stemming from proof mining, we extract a uniform rate of convergence for the successive distances between the lion and the man. As a byproduct of our analysis, we study the relation among different convexity properties in the setting of geodesic spaces.

Keywords

lion and man game, rate of convergence, uniform convexity, betweenness property

Mathematical Subject Classification 2010

Primary: 49N75, 91A24

Secondary: 03F10, 53C23

Milestones

Received: 8 August 2019

Revised: 18 May 2020

Accepted: 6 June 2020

Published: 26 January 2021