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

TL;DR

This paper introduces a uniform betweenness property in metric spaces and applies it to analyze a pursuit-evasion game, proving the lion's victory in convex domains and deriving convergence rates.

Contribution

It establishes a new geometric property and demonstrates its role in the quantitative analysis of pursuit-evasion games, linking convexity and convergence rates.

Findings

Lion always wins in uniformly convex bounded domains

Derived a uniform rate of convergence for the game

Explored relations among convexity properties in geodesic spaces

Abstract

In this paper 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 $\varepsilon$-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.