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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 Analytical inequalities involving real functions 26A51 Convexity, generalizations (one real variable) 28C20 Set functions and measures and integrals in infinite-dimensional spaces
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##### References:
 [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) · 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) [6] Brascamp, H. J.; Lieb, E. H.: Some inequalities for Gaussian measures. Functional integral and its applications (1975) · 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. Thesis (November 1973) · 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) · Zbl 0297.60004 [11] Borell, C.: Convex set functions. Period. math. Hungar. 6, 111-136 (1975) · Zbl 0307.28009