A note on Santaló inequality for the polarity transform and its reverse

Shiri Artstein-Avidan, Boaz A. Slomka

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We prove a Santaló and a reverse Santaló inequality for the class consisting of even log-concave functions attaining their maximal value 1 at the origin, also called even geometric log-concave functions. We prove that there exist universal numerical constants c,C > 0 such that for any even geometric log-concave function f = e−ϕ, (equation found) where Bn2 is the Euclidean unit ball of ℝn and ϕ° is the polar function of ϕ (not the Legendre transform!), a transform which was recently rediscovered by Artstein-Avidan and Milman and is defined below. The bounds are sharp up to the optimal constants c,C.

Original languageEnglish
Pages (from-to)1693-1704
Number of pages12
JournalProceedings of the American Mathematical Society
Issue number4
StatePublished - 2015
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2014 American Mathematical Society.


  • Log-concave function
  • Polarity transform
  • Santaló and reverse Santaló inequality


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