Decompositions of zonoids

Abstract

A zonoid is the range of a non atomic vector measure μ into ℝⁿ. We examine, by example, a result of Bolker: “Every zonoid K that is not a line segment is decomposable: i.e., is the sum of two sets, neither of which is a translate of a homothetic copy of K.” In particular we consider the zonoid K that is the closed unit (Euclidean) disk in ℝ² and show that a “natural decomposition” of K into a sum of two elliptic disks (of a certain kind) is not possible. Using a μ whose range is K, we give examples of decompositions of K into a sum of two disks such that these are ranges of restrictions of μ to a set A and its complement A ^c. This illustrates and complements a result of Rodriguez-Piazza. We also show that K is decomposable in the sense of Bolker.