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Functional Inequalities related to the Rogers-Shephard Inequality

Published online by Cambridge University Press:  21 December 2009

Andrea Colesanti
Affiliation:
Dipartimento di Matematica “U. Dini”, viale Morgagni 67/A, 50134 Firenze, Italy. E-mail: [email protected]
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Abstract

In this paper a notion of difference function Δf is introduced for real-valued, non-negative and log-concave functions f defined in Rn. The difference function represents a functional analogue of the difference body K + (−K) of a convex body K. The main result is a sharp inequality which bounds the integral of Δf from above in terms of the integral of f. Equality conditions are characterized. The investigation is extended to an analogous notion of difference function for α-concave functions, with α < 0. In this case also an upper bound for the integral of the α-difference function of f in terms of the integral of f is proved. The bound is sharp in the case α = −∞ and in the one-dimensional case.

Type
Research Article
Copyright
Copyright © University College London 2006

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References

1Artstein, S., Klartag, B. and Milman, V., The Santaló point of a function, and a functional form of Santaló inequality. Mathematika 51 (2004), 3348.CrossRefGoogle Scholar
2Ball, K. M., PhD dissertation, University of Cambridge (1987).Google Scholar
3Bonnesen, T. and Fenchel, W., Theory of Convex Bodies (translated from the German edition, 1934). BCS Associates (Moscow, Idaho, 1987).Google Scholar
4Brascamp, H. and Lieb, E., On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to diffusion equation. J. Funct. Anal. 22 (1976), 366389.CrossRefGoogle Scholar
5Fradelizi, M. and Meyer, M., Some functional forms of Santaló inequality. Preprint (2005).CrossRefGoogle Scholar
6Gardner, R. J., The Brunn-Minkowski inequality. Bull. Amer. Math. Soc. (N.S.) 39 (2002), 355405.CrossRefGoogle Scholar
7Klartag, B. and Milman, V., Geometry of log-concave functions and measures. Geom. Dedicata 112 (2005), 169182.CrossRefGoogle Scholar
8Lutwak, E., The Brunn-Minkowski-Firey theory I: mixed volumes and the Minkowski problem. J. Differential Geom. 38 (1993), 131150.CrossRefGoogle Scholar
9Rockafellar, R. T., Convex Analysis. Princeton University Press (Princeton, 1970).CrossRefGoogle Scholar
10Rogers, C. A. and Shephard, G. C., The difference body of a convex body. Arch. Math. 8 (1957), 220233.CrossRefGoogle Scholar
11Rogers, C. A. and Shephard, G. C., Convex bodies associated with a given convex body. J. Lond. Math. Soc. 33 (1958), 270281.CrossRefGoogle Scholar
12Schneider, R., Convex Bodies: the Brunn-Minkowski Theory. Encyclopedia of Mathematics and its Applications 44, Cambridge University Press (Cambridge, 1993).CrossRefGoogle Scholar