The case of equality for an inverse Santal functional inequality

M. Fradelizi, Y. Gordon, M. Meyer, S. Reisner

Research output: Contribution to journalArticlepeer-review


Let f : ℝn → ℝ+ be a log-concave function such that (x1, . . . , xn) = f (|x1|, . . . , |xn|) for every (x1, . . . .,xn) ∈ ℝn and o < ∫ℝn < + ∞.. It was proved in [4] and [5] iffor y ∈ ℝn,f* Fradelizi, Meyer, Positivity 12: 407420, 2008 and Fradelizi(y): = inf x∈ℝn e-(x,y)/f(x) then P (f) := ∫ℝn f(x) dx ∫ℝn f*(y) dy ≥ 4n, We characterize here the case of equality: one has P(f) = 4n4n if and only if can be written as f(x1, x2) e-||x1||K1 1K2 (x2), where and K1, ⊂ ℝn1 and K2 ⊂ ℝn2, n1 + n2 = n, are unconditional convex bodies such that P(Ki): = |Ki |Ki*| = 4ni/n i!,i = 1, 2, where Ki* denotes the polar of Ki. These last bodies were characterized in [7] and [8]

Original languageEnglish
Pages (from-to)621-630
Number of pages10
JournalAdvances in Geometry
Issue number4
StatePublished - Oct 2010


  • Inverse Santal inequality
  • Legendre transform
  • Log-concave functions
  • Mahler conjecture

ASJC Scopus subject areas

  • Geometry and Topology


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