Grünbaum's inequality for projections

M. Stephen*, N. Zhang

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


Let K be a convex body in Rn whose centroid is at the origin, let E∈G(n,k) be a subspace, and let ξ∈Sn−1. We find the best constant c=(kn+1)k so that |(K|E)∩ξ+|k≥c|K|E|k, and completely determine the minimizer. Here, |⋅|k is k-dimensional volume, K|E is the projection of K onto E, and ξ+={x∈Rn:〈x,ξ〉≥0}. Our result generalizes both Grünbaum's inequality, and an old inequality of Minkowski and Radon.

Original languageEnglish
Pages (from-to)2628-2640
Number of pages13
JournalJournal of Functional Analysis
Issue number6
StatePublished - 15 Mar 2017
Externally publishedYes


  • Centroid
  • Convex bodies
  • Grünbaum's inequality
  • Projections


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