Approximations of convex bodies by measure-generated sets

Han Huang, Boaz A. Slomka

Research output: Contribution to journalArticlepeer-review


Given a Borel measure μ on Rn, we define a convex set by M(μ)=⋃0≤f≤1,∫Rnfdμ=1{∫Rnyf(y)dμ(y)},where the union is taken over μ-measurable functions f: Rn→ [0 , 1] such that ∫Rnfdμ=1 and ∫Rnyf(y)dμ(y) exists. We study the properties of these measure-generated sets, and use them to investigate natural variations of problems of approximation of general convex bodies by polytopes with as few vertices as possible. In particular, we study an extension of the vertex index which was introduced by Bezdek and Litvak. As an application, we prove that for any non-degenerate probability measure μ, one has the lower bound ∫Rn∥x∥Z1(μ)dμ(x)≥cn,where Z1(μ) is the L1-centroid body of μ.

Original languageEnglish
Pages (from-to)173-196
Number of pages24
JournalGeometriae Dedicata
Issue number1
StatePublished - 1 Jun 2019
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2018, Springer Nature B.V.


  • Approximation
  • Banach–Mazur distance
  • Centroid bodies
  • Convex bodies
  • Vertex index


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