Concavity of powers of a convolution

Concavity of powers of a convolution
HTML articles powered by AMS MathViewer

Proc. Amer. Math. Soc. 82 (1981), 503-504

DOI: https://doi.org/10.1090/S0002-9939-1981-0612750-0

PDF | Request permission

A short geometric proof of a result of Brascamp and Lieb on concavity of powers of a convolution is given.

References

C. Borell, Convex set functions in $d$-space, Period. Math. Hungar. 6 (1975), no. 2, 111–136. MR 404559, DOI 10.1007/BF02018814
Herm Jan Brascamp and Elliott H. Lieb, On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, J. Functional Analysis 22 (1976), no. 4, 366–389. MR 0450480, DOI 10.1016/0022-1236(76)90004-5
Ju. S. Davidovič, B. I. Korenbljum, and B. I. Hacet, A certain property of logarithmically concave functions, Dokl. Akad. Nauk SSSR 185 (1969), 1215–1218 (Russian). MR 0241584
H. G. Eggleston, Convexity, Cambridge Tracts in Mathematics and Mathematical Physics, No. 47, Cambridge University Press, New York, 1958. MR 0124813, DOI 10.1017/CBO9780511566172
I. A. Ibragimov, On the composition of unimodal distributions, Teor. Veroyatnost. i Primenen. 1 (1956), 283–288 (Russian, with English summary). MR 0087249
C. G. Lekkerkerker, A property of logarithmic concave functions. I, II, Indag. Math. 15 (1953), 505–513, 514–521. Nederl. Akad. Wetensch. Proc. Ser. A 56. MR 0059977, DOI 10.1016/S1385-7258(53)50063-0
András Prékopa, On logarithmic concave measures and functions, Acta Sci. Math. (Szeged) 34 (1973), 335–343. MR 404557
Yosef Rinott, On convexity of measures, Ann. Probability 4 (1976), no. 6, 1020–1026. MR 428540, DOI 10.1214/aop/1176995947