Non-mutual captures in cyclic pursuit

In cyclic pursuit n bugs chase each other in cyclic order, each moving at u nit speed. Mathematical problems and puzzles of pursuit, and cyclic pursuit in particular, have attracted interest for many years. In 1971 Klamkin and Newman [17] showed that if n = 3 and the initial positions of the bugs are not collinear, then all three bugs capture their prey simultaneously, i.e. , no bug captures its prey prior to the moment when the pursuit collapses t o a single point. They asked whether the result generalizes to more bugs. B ehroozi and Gagnon [4] showed that it does generalize to n = 3 if the bugs' initial positions form a convex polygon. In this paper we resolve the gene ral question in k dimensions: It is possible for bugs to capture their prey without all bugs simultaneously doing so even for non-collinear initial po sitions. The set of initial conditions which give rise to non-mutual captur es is, however, a sub-manifold in the manifold of all possible initial cond itions. Hence, if the initial positions are picked randomly according to a smooth probability distribution, then the probability that a non-mutual cap ture will occur is zero.