Geometric inequalities for a class of exponential measures

Koenig, Hermann; Tomczak-Jaegermann, Nicole

doi:10.1090/S0002-9939-04-07862-1

Geometric inequalities for a class of exponential measures
HTML articles powered by AMS MathViewer

by Hermann Koenig and Nicole Tomczak-Jaegermann PDF

Proc. Amer. Math. Soc. 133 (2005), 1213-1221 Request permission

Abstract:

Using $M$-ellipsoids we prove versions of the inverse Santaló inequality and the inverse Brunn-Minkowski inequality for a general class of measures replacing the usual volume on $\mathbb {R}^n$. This class contains in particular the Gaussian measure on $\mathbb {R}^n$.

References

J. Bourgain and V. D. Milman, New volume ratio properties for convex symmetric bodies in $\textbf {R}^n$, Invent. Math. 88 (1987), no. 2, 319–340. MR 880954, DOI 10.1007/BF01388911
Dario Cordero-Erausquin, Santaló’s inequality on $\Bbb C^n$ by complex interpolation, C. R. Math. Acad. Sci. Paris 334 (2002), no. 9, 767–772 (English, with English and French summaries). MR 1905037, DOI 10.1016/S1631-073X(02)02328-2
D. Cordero-Erausquin, M. Fradelizi and B. Maurey, The (B) conjecture for the Gaussian measure of dilates of symmetric convex sets and related problems. preprint.
R. J. Gardner, The Brunn-Minkowski inequality, Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 3, 355–405. MR 1898210, DOI 10.1090/S0273-0979-02-00941-2
Gilles Hargé, Une inégalité de décorrélation pour la mesure gaussienne, C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), no. 11, 1325–1328 (French, with English and French summaries). MR 1649146, DOI 10.1016/S0764-4442(98)80188-2
Vitali D. Milman, Inégalité de Brunn-Minkowski inverse et applications à la théorie locale des espaces normés, C. R. Acad. Sci. Paris Sér. I Math. 302 (1986), no. 1, 25–28 (English, with French summary). MR 827101
Gilles Pisier, The volume of convex bodies and Banach space geometry, Cambridge Tracts in Mathematics, vol. 94, Cambridge University Press, Cambridge, 1989. MR 1036275, DOI 10.1017/CBO9780511662454
G. Schechtman, Th. Schlumprecht, and J. Zinn, On the Gaussian measure of the intersection, Ann. Probab. 26 (1998), no. 1, 346–357. MR 1617052, DOI 10.1214/aop/1022855422