Abstract
In his paper [M] B. Maurey gave a very simple proof of the gaussian deviation inequality, based on an inequality of A. Prekopa [Pr] and L. Leindler [L]. Motivated by Talagrand’s isoperimetric inequality for the cube (cf. [T.3]) Maurey defined a property which he called property (τ). He proved that in the gaussian case this property is a consequence of the Prekopa-Leindler inequality. This method can be easily extended to give a concentration of measure phenomenon for a particular measure associated with uniformly convex spaces whose modulus of convexity satisfies a certain condition. In [Pi] G. Pisier proved that this condition can be guaranteed after renorming. Prom this we deduce the concentration of measure result of M. Gromov and V.D. Milman (cf. [G.M.2]) for uniformly convex bodies, whose modulus of convexity satisfies the condition mentioned above.
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References
K. Ball, unpublished.
M.M. Day. Some characterizations of inner product spaces, Trans. Amer. Math. Soc. 62 (1947), 320–337.
M. Gromov and V.D. Milman. Brunn’s theorem and a concentration of measure phenomenon for convex symmetric bodies, GAFA 1983/84.
M. Gromov and V.D. Milman. Generalization of the spherical isoperimetric inequality to uniformly convex Banach spaces, Comp. Math. 62 (1987), 263–282.
M. Ledoux and M. Talagrand, Probability in Banach Spaces, Springer, 1991.
L. Leindler. On a certain converse of Hölder’s inequality II, Acta Sci. Math. 33 (1972), 217–223.
J. Lindenstrauss and L. Tzafriri. Classical Banach Spaces II. Springer
M. Meyer and A. Pajor. Sections of the unit ball of ℓ n p . Journal of Funct. Anal. 80 (1988), 109–123.
V.D. Milman and A. Pajor. Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n dimensional space. GAFA 87/88, SLNM 1376, 64–104.
M. Meyer and S. Reisner. A geometric property of the boundary of symmetric convex bodies and convexity of notation surfaces, Geom. Ded. 37 (1991), 327–337.
B. Maurey. Some deviation inequalities, GAFA 1.2 (1991), 188–197.
A. Pajor. Notes
G. Pisier. Martingales with values in uniformly convex spaces, Israel. J. Math. 20, 3–4 (1975),
G. Pisier. Martingales with values in uniformly convex spaces, Israel. J. Math. 20, 326–350 (1975).
A. Prekopa. On logarithmically concave measures and functions, Acta Sci. Math 34 (1973) 335–343.
C. Schütt and E. Werner. The convex floating body, Math. Scand. 66 (1990), 275–290.
G. Schechtman and J. Zinn. On the volume of the intersection of two L n p balls, Proc. Amer. Math. Soc. 110 (1990), 217–224.
M. Talagrand, A new isoperimetric inequality and the concentration of measure phenomenon, GAFA 1469 (1991), 94–124.
M. Talagrand. The supremum of some canonical processes, Manuscript.
M. Talagrand. An Isoperimetric Theorem on the Cube and the Khintchine-Kahane Inequalities, Proc. Amer. Math. Soc. 104 (1988), 905–909.
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© 1995 Birkhäuser Verlag Basel/Switzerland
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Schmuckenschläger, M. (1995). A Concentration of Measure Phenomenon on Uniformly Convex Bodies. In: Lindenstrauss, J., Milman, V. (eds) Geometric Aspects of Functional Analysis. Operator Theory Advances and Applications, vol 77. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9090-8_23
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