Lift zonoid and barycentric representation on a Banach space with a cylinder measure

TL;DR

This paper extends the lift zonoid concept from finite-dimensional spaces to infinite-dimensional Banach spaces with cylindrical measures, providing a barycentric representation of interior points of convex hulls.

Contribution

It introduces a novel infinite-dimensional generalization of the lift zonoid concept for cylindrical measures, enabling barycentric representations in Banach spaces.

Findings

Established a one-to-one barycentric representation in finite dimensions.

Generalized the lift zonoid concept to infinite-dimensional Banach spaces.

Proved the representation for cylindrical probability measures.

Abstract

We show that the lift zonoid concept for a probability measure on R^d, introduced in (Koshevoy and Mosler, 1997), leads naturally to a one-to one representation of any interior point of the convex hull of the support of a continuous measure as the barycenter w.r.t. to this measure of either of a half-space, or the whole space. We prove the infinite-dimensional generalization of this representation, which is based on the extension of the lift-zonoid concept for a cylindrical probability measure.