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

Abstract

We show that the lift zonoid concept for a probability measure on \( {{\mathbb{R}}^d} \), introduced in [G.A. Koshevoy and K. Mosler, Zonoid trimming for multivariate distributions, Ann. Stat., 25(5):1998–2017, 1997], naturally leads to a oneto-one representation of any interior point of the convex hull of the support of a continuous measure as the barycenter w.r.t. this measure of either a half-space or the whole space. We prove an infinite-dimensional generalization of this representation, which is based on the extension of the concept of lift zonoid for a cylindrical probability measure.

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Correspondence to Alexei M. Kulik.

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Kulik, A.M., Tymoshkevych, T.D. Lift zonoid and barycentric representation on a Banach space with a cylinder measure. Lith Math J 53, 181–195 (2013). https://doi.org/10.1007/s10986-013-9202-z

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Keywords

  • zonoid
  • lift zonoid
  • cylinder measure
  • barycentric representation

MSC

  • primary 60D05
  • secondary 28C20