Spline subdivision schemes for convex compact sets

The application of spline subdivision schemes to data consisting of convex compact sets, with addition replaced by Minkowski sums of sets, is investigated. These methods generate in the limit set-valued functions, which can be expressed explicitly in terms of linear combinations of integer shifts of B-splines with the initial data as coefficients. The subdivision techniques are used to conclude that these limit set-valued spline functions have shape-preserving properties similar to those of the usual spline functions. This extension of subdivision methods from the scalar setting to the set-valued case has application in the approximate reconstruction of 3-D bodies from finite collections of their parallel cross-sections.