  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.
