In a pursuit-evasion game, a team of pursuers attempt to capture an evader. The players alternate turns, move with equal speed, and have full information about the state of the game. We consider the most restictive capture condition: a pursuer must become colocated with the evader to win the game. We prove two general results about pursuit-evasion games in topological spaces. First, we show that one pursuer has a winning strategy in any CAT(0) space under this restrictive capture criterion. This complements a result of Alexander, Bishop and Ghrist, who provide a winning strategy for a game with positive capture radius. Second, we consider the game played in a compact domain in Euclidean two-space with piecewise analytic boundary and arbitrary Euler characteristic. We show that three pursuers always have a winning strategy by extending recent work of Bhadauria, Klein, Isler and Suri from polygonal environments to our more general setting.
A pursue-and-evasion game is analyzed, including almost optimal bounds on the number of moves needed to win. Frontispiece Preface Foreword 1. Introduction to A Mathematician's Miscellany 2. Mathematics with minimum 'Raw Material' 3. From the mathematical tripos 4. Cross-purposes, unconscious assumptions, howlers, misprints, etc 5. The zoo 6. Ballistics 7. The dilemma of probability theory 8. From Fermat's last theorem to the abolition of capital punishment 9. A mathematical education 10. Review of Ramanujan's collected papers 11. Large numbers 12. Lion and man 13. People 14. Academic life 15. Odds and ends 16. Newton and the attraction of a sphere 17. The discovery of Neptune 18. The Adams-Airy affair 19. The mathematician's art of work.