In this paper, a new family of pursuit strategies is presented for a fundamental pursuit-evasion game: the lion and man problem. The classic formulation due to David Gale is considered: the players move alternatively in a discrete-time setting and are constrained to remain in the non-negative quadrant of the plane. The proposed solution relies on the stepwise optimization of a suitable functional, whose generality provides an appealing degree of freedom for customization within specific pursuit problems and bounded environments. Numerical simulations show the benefits of the new approach with respect to existing pursuit strategies.