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On extensions of the Brunn-Minkowski and Prekopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation. (English) Zbl 0334.26009

MSC:
26D20 Other analytical inequalities
26A51 Convexity of real functions in one variable, generalizations
28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
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[1] Lusternik, L, Die brunn-minkowskische ungleichung für beliebige messbare mengen, C. R. dokl. acad. sci. URSS no. 3, 8, 55-58, (1935) · Zbl 0012.27203
[2] Federer, M, Geometric measure theory, (1969), Springer New York · Zbl 0176.00801
[3] Prékopa, A, Logarithmic concave measures with application to stochastic programming, Acta sci. math. (Szeged), 32, 301-315, (1971) · Zbl 0235.90044
[4] Leindler, L, On a certain converse of Hölder’s inequality II, Acta sci. math. (Szeged), 33, 217-223, (1972) · Zbl 0245.26011
[5] Prékopa, A, On logarithmic concave measures and functions, Acta sci. math. (Szeged), 34, 335-343, (1973) · Zbl 0264.90038
[6] Brascamp, H.J; Lieb, E.H, Some inequalities for Gaussian measures, () · Zbl 0348.26011
[7] Brascamp, H.J; Lieb, E.H, Best constants in Young’s inequality, its converse and its generalization to more than three functions, Advances in math., 20, (1976) · Zbl 0339.26020
[8] Rinott, Y, On convexity of measures, (), to appear · Zbl 0347.60003
[9] Simon, B; Høegh-Krohn, R, Hypercontractive semigroups and two-dimensional self-coupled Bose fields, J. functional analysis, 9, 121-180, (1972) · Zbl 0241.47029
[10] Borell, C, Convex measures on locally convex spaces, Ark. mat., 12, 239-252, (1974), Note added in proof. After this paper was submitted for publication we discovered that Corollary 3.4 and its converse were proved by C. Borell: · Zbl 0297.60004
[11] Borell, C, Convex set functions, Period. math. hungar., 6, 111-136, (1975) · Zbl 0274.28009
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