Hostname: page-component-745bb68f8f-grxwn Total loading time: 0 Render date: 2025-01-11T02:06:34.668Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: General convexity

Published online by Cambridge University Press:  26 February 2010

S. Artstein-Avidan ,
B. Klartag  and
V. Milman
S. Artstein-Avidan
Affiliation:
Department of Mathematics, Princeton University, Princeton NJ 08544–1000, USA, and School of Mathematics, Institute for Advanced Study, 1 Einstein Drive, Princeton NJ, 08450, USA E-mail: [email protected] and [email protected]
B. Klartag
Affiliation:
School of Mathematics, Institute for Advanced Study, 1 Einstein Drive, Princeton NJ, 08450, USA E-mail: [email protected]
V. Milman
Affiliation:
School of Mathematical Science, Tel Aviv University, Ramat Aviv, Tel Aviv, 69978, Israel. E-mail: [email protected]
Get access

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 f(x) = f(x−a) such that

MSC classification

Secondary: 52A40: Inequalities and extremum problems
Type
Research Article
Information
Mathematika , Volume 51 , Issue 1-2 , December 2004 , pp. 33 - 48
Copyright
Copyright © University College London 2004

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[Ar]Arnold, V. I.. Mathematical Methods of Classical Mechanics (Translated from the Russian by Vogtmann, K. and Weinstein, A., second edition). Graduate Texts in Mathematics, 60. Springer-Verlag (New York, 1989).Google Scholar
[Bal]Ball, K.. PhD dissertation. University of Cambridge (1987).Google Scholar
[Ba2]Ball, K.. Logarithmically concave functions and sections of convex sets in ℝ. Studia Math., 88 (1988), 69–84.CrossRefGoogle Scholar
[Bar]Barthe, F.. On a reverse form of the Brascamp-Lieb inequality. Invent. Math., 134 (1988). 335–361.CrossRefGoogle Scholar
[Bo]Borell, C.. Convex set functions in d-space. Period. Math. Hungar., 6 (1975). 111–136.CrossRefGoogle Scholar
[Br]Brunn, H.. Referat Ober eine Arbeit: Exacte Grundlagen für eincr Theorie der Ovale (Bayer, S. B., ed.), Akad. Wiss., (1894), 93–111.Google Scholar
[K]Klartag, B.. An isomorphic version of the slicing problem, J. Fund. Anal., 218 (2005). 372–394.CrossRefGoogle Scholar
[KM]Klartag, B. and Milman, V. D.. Geometry of log-concave functions and measures. Geom. Dedicata, 112 (2005), 169–182.CrossRefGoogle Scholar
[LO]Linnik, J. V. and Ostrovis'kiĭ, I. V., Decomposition of random variables and vectors (translated from the Russian). Translations of Mathematical Monographs. Vol. 48. American Mathematical Society, (Providence, R. I., 1977).Google Scholar
[Mau]Maurey, B.. Some deviation inequalities. Geom. Fund. Anal., 1 (1991), 188197.CrossRefGoogle Scholar
[MeP1]Meyer, M. and Pajor, A.. On Santalo's inequality. Geometric Aspects of Functional Analysis (1987–88), Lecture Notes in Math., 1376, Springer (Berlin, 1989). 261–263.CrossRefGoogle Scholar
[MeP2]Meyer, M. and Pajor, A.. On the Blaschke-Santalo inequality. Arch. Math., 55 (1990). 82–93.CrossRefGoogle Scholar
[Mi1]Milman, V. D.. A certain transformation of convex functions and a duality of the β and δ-characteristics of a B-space. Dokl Akad, Nauk SSSR (1969), 33–35; Soviet Math. Dok., 10 (1969), 789–792.Google Scholar
[Mi2]Milman, V. D.. Duality of some geometric characteristics of Banach spaces. Tear. Funksii. Funksional Analiz i Prilozhen, 18 (1973), 120–137.Google Scholar
[P]Pisier, G.. The Volume of Convex Bodies and Banach Space Geometry. Cambridge Tracts in Mathematics, 94. Cambridge University Press (Cambrige, 1989).CrossRefGoogle Scholar
[R]Rockafellar, R. T.. Convex Analysis. Princeton Mathematical Series, No. 28, Princeton University Press (Princeton, N.J., 1970).Google Scholar
[Sa]Santaló, L. A.. An affine invariant for convex bodies of n-dimensional space. (Spanish). Portugaliae Math., 8 (1949) 155–161.Google Scholar
[T]Talagrand, M.. A new isoperimetric inequality and the concentration of measure phenomenon. Geom. Aspects of Fund. Anual. (1989-90), Lecture Notes in Math..1469, Springer (Berlin, 1991), 94–124.Google Scholar