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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: 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, (Arthurs, A. M., Functional Integral and its Applications (1975), Clarendon Press: Clarendon Press Oxford) · 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), Weizmann Institute: Weizmann Institute Rehovot, Israel), 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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