We introduce simple tools from geometric convexity to analyze capture-type (or "Lion and Man'') pursuit problems in unbounded domains. The main result is a necessary and sufficient condition for eventual capture in equal-speed discrete-time multi-pursuer capture games on convex Euclidean domains of arbitrary dimension and shape. This condition is presented in terms of recession sets in unit tangent spheres. The chief difficulties lie in utilizing the boundary of the domain as a constraint on the evader's escape route. We also show that these convex-geometric techniques provide sufficient criteria for pursuit problems in non-convex domains with a convex decomposition.
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S. Alexander, R. Bishop, Robert Ghrist, Capture pursuit games on unbounded domains. Enseign. Math. 55 (2009), no. 1, pp. 103–125DOI 10.4171/LEM/55-1-5