A note on Santaló inequality for the polarity transform and its reverse
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- by Shiri Artstein-Avidan and Boaz A. Slomka PDF
- Proc. Amer. Math. Soc. 143 (2015), 1693-1704 Request permission
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 $f=e^{-\varphi }$, \[ c^{n}\cdot \left |B_{2}^{n}\right |^{2}\le \int _{\mathbb {R}^{n}}e^{-\varphi }\int _{\mathbb {R}^{n}}e^{-\varphi ^{\circ }} \le \left (\left |B_{2}^{n}\right |n!\right )^{2}\left (1+\frac {C}{n}\right ) \] 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 $c,C$.References
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Additional Information
- Shiri Artstein-Avidan
- Affiliation: School of Mathematical Science, Tel-Aviv University, Ramat Aviv, Tel Aviv, 69978, Israel
- MR Author ID: 708154
- 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
- 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
- © Copyright 2014 American Mathematical Society
- 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
- MathSciNet review: 3314082