The santaló point of a function, and a functional form of the santaló inequality

S. Artstein-Avidan*, B. Klartag, V. Milman

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

118 Scopus citations

Abstract

Let ℒ(f) denote the Legendre transform of a function f: ℝn → ℝ. A theorem of K. Ball about even functions is generalized, and it is proved that, for any measurable function f ≥ 0, there exists a translation f̃(x) = f(x - a) such that ∫ℝne -f̃ℝne-ℒ(f̃)≤ (2π)n. (1) If a is selected so as to minimize the left side of (1), then equality in (1) is satisfied if and only if e-f is proportional to the distribution of a Gaussian random variable. This inequality immediately implies the Santaló inequality for convex bodies, as well as a new concentration inequality for the Gaussian measure.

Original languageEnglish
Pages (from-to)33-48
Number of pages16
JournalMathematika
Volume51
Issue number1-2
DOIs
StatePublished - 2004

Funding

FundersFunder number
National Science FoundationDMS-0111298
Bonfils-Stanton Foundation

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