K-transversals of Parallel Convex Sets

R d can be divided into a union of parallel d k ats of the form x g x g xk gk where the gi are constant Let C be a family of parallel d k dimensional convex sets meaning that each is contained in one of the above parallel d k ats We give a parameterization of the set of k ats in R such that the set of k ats which intersect in a point any set c C is convex Parameterizing the lines in R through horizontal convex sets as convex sets has applications to medical imaging and interesting connections with recent work on light eld rendering in computer graphics The general case is useful for tting k ats to points in R The following easy reduction is well known Let C be a nite set of parallel line segments in R d We want to nd a d transversal for C that is a hyperplane intersecting every segment in C Such a hyperplane has to pass below the upper endpoint of each segment and above the lower endpoint In the dual the endpoints correspond to linear halfspaces and the intersection of these halfspaces corresponds to the set of hyperplane transversals of the parallel segments in the primal So the problem is solved by linear programming in dimension d in linear time if d is xed Here we give the appropriate generalization of this observation for k transversals for k d A k transversal of a family of sets C is a k at that is a k dimensional a ne subspace intersecting every set in C Figure shows the case k d Figure The set of lines intersecting all members of a family of parallel polygons can be represented as a convex set A family C of d k dimensional sets in R are parallel if they can be rotated so that each set c C lies in a d k at x g x g xk gk where g gk are constants and x xk are the rst k coordinates of a point x R d From now on we will just assume that C is so rotated We say that a k at y intersects a set c C non degenerately if y c consists of a single point a k at in general position intersects a set c C non degenerately if at all Our Main Theorem gives a parameterization under which the k ats intersecting non degenerately any member of C form a convex set in R k d k This result is a simple algebraic consequence of adopting the right parameterization of k ats in R But it has both mathematical and practical implications