Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 
 
 

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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$.

References [Enhancements On Off] (What's this?)

  • [1] S. Artstein-Avidan, B. Klartag, and V. Milman, The Santaló point of a function, and a functional form of the Santaló inequality, Mathematika 51 (2004), no. 1-2, 33-48 (2005). MR 2220210 (2007a:52008), https://doi.org/10.1112/S0025579300015497
  • [2] S. Artstein-Avidan and V. Milman, A characterization of the support map, Adv. Math. 223 (2010), no. 1, 379-391. MR 2563222 (2011g:52002), https://doi.org/10.1016/j.aim.2009.07.020
  • [3] S. Artstein-Avidan and V. Milman, Hidden structures in the class of convex functions and a new duality transform, J. Eur. Math. Soc. (JEMS) 13 (2011), no. 4, 975-1004. MR 2800482 (2012h:49069), https://doi.org/10.4171/JEMS/273
  • [4] K. M. Ball, PhD dissertation, Cambridge.
  • [5] J. Bourgain and V. D. Milman, New volume ratio properties for convex symmetric bodies in $ {\bf R}^n$, Invent. Math. 88 (1987), no. 2, 319-340. MR 880954 (88f:52013), https://doi.org/10.1007/BF01388911
  • [6] M. Fradelizi and M. Meyer, Some functional forms of Blaschke-Santaló inequality, Math. Z. 256 (2007), no. 2, 379-395. MR 2289879 (2008c:52013), https://doi.org/10.1007/s00209-006-0078-z
  • [7] M. Fradelizi and M. Meyer, Increasing functions and inverse Santaló inequality for unconditional functions, Positivity 12 (2008), no. 3, 407-420. MR 2421143 (2009e:26025), https://doi.org/10.1007/s11117-007-2145-z
  • [8] B. Klartag and V. D. Milman, Geometry of log-concave functions and measures, Geom. Dedicata 112 (2005), 169-182. MR 2163897 (2006d:52004), https://doi.org/10.1007/s10711-004-2462-3
  • [9] G. Kuperberg, From the Mahler conjecture to Gauss linking integrals, Geom. Funct. Anal. 18 (2008), no. 3, 870-892. MR 2438998 (2009i:52005), https://doi.org/10.1007/s00039-008-0669-4
  • [10] J. Lehec, A direct proof of the functional Santaló inequality, C. R. Math. Acad. Sci. Paris 347 (2009), no. 1-2, 55-58 (English, with English and French summaries). MR 2536749 (2010h:26027), https://doi.org/10.1016/j.crma.2008.11.015
  • [11] J. Lehec, Partitions and functional Santaló inequalities, Arch. Math. (Basel) 92 (2009), no. 1, 89-94. MR 2471991 (2010a:39017), https://doi.org/10.1007/s00013-008-3014-0
  • [12] M. Meyer and A. Pajor, On the Blaschke-Santaló inequality, Arch. Math. (Basel) 55 (1990), no. 1, 82-93. MR 1059519 (92b:52013), https://doi.org/10.1007/BF01199119
  • [13] V. Milman and L. Rotem, Mixed integrals and related inequalities, J. Funct. Anal. 264 (2013), no. 2, 570-604. MR 2997392, https://doi.org/10.1016/j.jfa.2012.10.019
  • [14] F. Nazarov, The Hörmander proof of the Bourgain-Milman theorem, Geometric aspects of functional analysis, Lecture Notes in Math., vol. 2050, Springer, Heidelberg, 2012, pp. 335-343. MR 2985302, https://doi.org/10.1007/978-3-642-29849-3_20
  • [15] R. Tyrrell Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. MR 0274683 (43 #445)
  • [16] L. Rotem, Characterization of self-polar convex functions, Bull. Sci. Math. 136 (2012), no. 7, 831-838. MR 2972564, https://doi.org/10.1016/j.bulsci.2012.03.003
  • [17] L. A. Santaló, An affine invariant for convex bodies of $ n$-dimensional space, Portugaliae Math. 8 (1949), 155-161 (Spanish). MR 0039293 (12,526f)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 52A41, 26A51, 46B10

Retrieve articles in all journals with MSC (2010): 52A41, 26A51, 46B10


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

American Mathematical Society