Grünbaum's inequality for sections

S. Myroshnychenko, M. Stephen*, N. Zhang

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


We show [Formula presented]≥([Formula presented])[Formula presented] for all k-dimensional subspaces E⊂Rn, θ∈E∩Sn−1, and all γ-concave functions f:Rn→[0,∞) with γ>0, 0<∫Rn f(x)dx<∞ and ∫Rn xf(x)dx at the origin o∈Rn. Here, θ+:={x:〈x,θ〉≥0}. As a consequence of this result, we get the following generalization of Grünbaum's inequality: [Formula presented]≥([Formula presented])k for all convex bodies K⊂Rn with centroid at the origin, k-dimensional subspaces E⊂Rn, and θ∈E∩Sn−1. The lower bounds in both of our inequalities are the best possible, and we discuss the equality conditions.

Original languageEnglish
Pages (from-to)2516-2537
Number of pages22
JournalJournal of Functional Analysis
Issue number9
StatePublished - 1 Nov 2018
Externally publishedYes


  • Centroid
  • Convex Bodies
  • Grünbaum's inequality
  • Sections


Dive into the research topics of 'Grünbaum's inequality for sections'. Together they form a unique fingerprint.

Cite this