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

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