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


Authors: Shiri Artstein-Avidan and Boaz A. Slomka
Journal: Proc. Amer. Math. Soc. 143 (2015), 1693-1704
MSC (2010): Primary 52A41, 26A51, 46B10
DOI: https://doi.org/10.1090/S0002-9939-2014-12390-2
Published electronically: December 9, 2014
MathSciNet review: 3314082
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Abstract: 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^{-\varphi }$,

$\displaystyle c^{n}\cdot \left \vert B_{2}^{n}\right \vert^{2}\le \int _{\mathb... ...(\left \vert B_{2}^{n}\right \vert n!\right )^{2}\left (1+\frac {C}{n}\right ) $

where $ B_{2}^{n}$ is the Euclidean unit ball of $ \mathbb{R}^{n}$ and $ \varphi ^{\circ }$ is the polar function of $ \varphi $ (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$.

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Additional Information

Shiri Artstein-Avidan
Affiliation: School of Mathematical Science, Tel-Aviv University, Ramat Aviv, Tel Aviv, 69978, Israel
Email: shiri@post.tau.ac.il

Boaz A. Slomka
Affiliation: School of Mathematical Science, Tel-Aviv University, Ramat Aviv, Tel Aviv, 69978, Israel
Email: boazslom@post.tau.ac.il

DOI: https://doi.org/10.1090/S0002-9939-2014-12390-2
Keywords: Santal\'o and reverse Santal\'o inequality, polarity transform, log-concave function
Received by editor(s): April 2, 2013
Received by editor(s) in revised form: June 12, 2013
Published electronically: December 9, 2014
Additional Notes: This work was supported by ISF grant No. 247/11.
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2014 American Mathematical Society