 
Summary: Ktransversals of parallel convex sets
Nina Amenta
Xerox PARC
3333 Coyote Hill Road
Palo Alto, CA 94304, USA
amenta@parc.xerox.com
May 6, 1996
Abstract
R d can be divided into a union of parallel (d\Gammak)flats of the form x 1 = g 1 ; x 2 = g 2 ; : : : x k =
g k , where the g i are constant. Let C be a family of parallel (d \Gamma k)dimensional convex
sets, meaning that each is contained in one of the above parallel (d \Gamma k)flats. We give a
parameterization of the set of kflats in R d , such that the set of kflats which intersect, in a
point, any set c 2 C, is convex. Parameterizing the lines in R 3 through horizontal convex
sets as convex sets has applications to medical imaging, and interesting connections with
recent work on light field rendering in computer graphics. The general case is useful for
fitting kflats to points in R d .
The following easy reduction is well known. Let C be a finite set of parallel line segments in
R d . We want to find a (d \Gamma 1)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
