Characterization of Optimal Trajectories in Double-struck capital R3

We characterize the set of all trajectories T of an object t moving in a given corridor Y that are furthest away from a family S = {S} of fixed unfriendly observers. Each observer is equipped with a convex open scanning cone K(S) with vertex S. There are constraints on the multiplicity of covering the corridor Y by the cones K and on the "thickness" of the cones. In addition, pairs S, S for which [S, S ]. (K(S) n K(S)) are not allowed. The original problem maxT min{d(t, S) : t. T, S. S}, where d(t, S) = t - S for t. K(S) and d(t, S) = +8 for t . K(S), is reduced to the problem of finding an optimal route in a directed graph whose vertices are closed disjoint subsets (boxes) from Y \ S K(S). Neighboring (adjacent) boxes are separated by some cone K(S). An edge is a part T (S) of a trajectory T that connects neighboring boxes and optimally intersects the cone K(S). The weight of an edge is the deviation of S from T (S). A route is optimal if it maximizes the minimum weight.