Pursuit and Evasion in Non-convex Domains of Arbitrary Dimensions

Most results in pursuit-evasion games apply only to planar domains or perhaps to higher-dimensional domains which must be convex. We introduce a very general set of techniques to generalize and extend certain results on simple pursuit to non-convex domains of arbitrary dimension which satisfy a coarse curvature condition (the CAT(0) condition). I. PURSUIT / EVASION There is a significant literature on pursuit-evasion games, with natural motivations coming from robotics [10], [14], [24]. Such games involve one or more evaders in a fixed domain being hunted by one or more pursuers who win the game if the appropriate capture criteria are satisfied. Such criteria may be physical capture (the pursuers move to where the evaders are located) [12], [13], [20] or visual capture (there is a line-of-sight between a pursuer and an evader) [10], [23]. The types of pursuit games are many and varied: continuous or discrete time, bounded or unbounded speed, and constraints on admissible acceleration, energy expenditure, strategy, and sensing. For a quick introduction to the literature on pursuit games, see, e.g., [15], [10]. This paper focuses on one particular variable in pursuit games: the geometry and topology of the domain on which the game is played. The vast majority of the known results on pursuit-evasion are dependent on having domains which are two-dimensional or, if higher-dimensional, then convex. There has of late been a limited number of results for pursuit games on surfaces of revolution [11], cones [18], and round spheres [16]. Our results are complementary to these, in the sense that we work with domains having dimension higher than two, without constraints on being smooth or a manifold. The principal contribution of this work is a significant extension of known results on convex or planar domains to domains of arbitrary dimension which satisfy a type of curvature constraint known as the CAT(0) condition. Roughly speaking, the CAT(0) condition is a measure of what triangles in a metric space (X, d) look like, and, in particular, how a triangle compares to a Euclidean triangle with the same three side lengths. A simple mnemonic for a CAT(0) space is that it is a metric space, all of whose geodesic triangles have an angle sum no greater than π. Examples of CAT(0) domains are numerous and include the following: 1) convex Euclidean domains; 2) simply-connected subsets of E; 3) simply-connected Riemannian manifolds with nonpositive sectional curvature; 4) smooth Euclidean domains with boundaries having no more than one non-convex direction at each point; 5) simply-connected piecewise-Euclidean cubical complexes with no positive discrete curvature at the vertices; 6) Euclidean rectangular prisms with certain cylindrical sets removed; 7) simply-connected unions of convex sets which have no triple intersections. Our goal in this paper is to motivate the adoption of CAT(0) techniques in this and other areas of robotics in which the generalization of results from 2-d to higher dimensions is problematic. Decades of work by geometers in CAT(0) and more general Alexandrov geometry (geometry of spaces of bounded curvature) forms a powerful set of tools which are not very visible outside of mathematics departments (see [4], [5] and §III below.). The proofs of pursuit/evasion results in this paper are all very simple and very short, given the appropriate standard results from comparison geometry. We extend results to CAT(0) spaces in a dimension-independent manner, and, often, examples which are of high dimension are no more difficult than those with dimension two: the same proofs cover all cases. After giving a motivational example of a simple pursuit problem in the plane (§II-B), we motivate the notion of comparison triangles, total curvature bounds, and their utility in simple pursuit problems. We then present a brief primer on CAT(0) geometry in §III, followed by a more technical result on growth rates of total curvature in §IV. These tools are used in §V to solve problems involving simple pursuit curves. We conclude this note with results on escape criteria (§VI), contrasts with the positive curvature case (§VII), and a series of remarks and open directions (§VIII). II. A QUICK SUMMARY OF PURSUIT