Zonoid

Summary

A zonoid is a convex body that is a limit, in the Hausdorff distance, of zonotopes, that is, of finite Minkowski sums of line segments. Zonoids are exactly the convex bodies whose support functions are integrals of the functions $x \mapsto \abs{x \cdot v}$, up to a translation. They appear in the geometry of finite-dimensional normed spaces and in the study of projections of convex bodies. By Lyapunov's convexity theorem, they also arise as the ranges of nonatomic vector-valued measures.