Vol. 310, No. 1, 2021

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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
Milestones
Received: 8 August 2019
Revised: 18 May 2020
Accepted: 6 June 2020
Published: 26 January 2021
Authors
Ulrich Kohlenbach
Department of Mathematics
Technische Universität Darmstadt
Darmstadt
Germany
Genaro López-Acedo
Department of Mathematical Analysis - IMUS
University of Seville
Sevilla
Spain
Adriana Nicolae
Department of Mathematics
Babes-Bolyai University
Cluj-Napoca
Romania