#### Vol. 310, No. 1, 2021

 Recent Issues Vol. 317: 1 Vol. 316: 1  2 Vol. 315: 1  2 Vol. 314: 1  2 Vol. 313: 1  2 Vol. 312: 1  2 Vol. 311: 1  2 Vol. 310: 1  2 Online Archive Volume: Issue:
 The Journal Subscriptions Editorial Board Officers Contacts Submission Guidelines Submission Form Policies for Authors ISSN: 1945-5844 (e-only) ISSN: 0030-8730 (print) Special Issues Author Index To Appear Other MSP Journals
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
##### 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