Escaping offline searchers and isoperimetric theorems

Given a set of searchers in the grid, whose search paths are known in advance, can a target that moves at the same speed as the searchers escape detection indefinitely? We study the number of searchers against which the target can still escape. This number is less than n in an n×n grid, since a row of searchers can sweep the allowed region.In an alternating-move-model where at each time searchers first move and then the target moves, we show that a target can always escape searchers and there is a strategy for searchers to catch the target. This improves a recent bound [A. Dumitrescu, I. Suzuki, P. Zylinski, Offline variants of the “lion and man” problem, in: SoCG 2007, Proc. 23rd Annual Symposium on Computational Geometry, ACM Press, 2007, pp. 102–111] in the simultaneous-move-model where at each time searchers and target moves simultaneously. We also prove similar bounds for the continuous analogue, as well as for searchers and targets moving with different speeds. In the proof, we use new isoperimetric theorems for subsets of the n×n grid and the n×n square, which is of independent interest.