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


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.)
Full Text: DOI


[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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