On the improvement of concavity of convex measures

Marsiglietti, Arnaud

doi:10.1090/proc/12694

On the improvement of concavity of convex measures
HTML articles powered by AMS MathViewer

by Arnaud Marsiglietti

Proc. Amer. Math. Soc. 144 (2016), 775-786

DOI: https://doi.org/10.1090/proc/12694

Published electronically: June 24, 2015

PDF | Request permission

Abstract:

We prove that a general class of measures, which includes $\log$-concave measures, is $\frac {1}{n}$-concave according to the terminology of Borell, with additional assumptions on the measures or on the sets, such as symmetries. This generalizes results of Gardner and Zvavitch published in 2010.

References

T. Bonnesen and W. Fenchel, Theory of convex bodies, BCS Associates, Moscow, ID, 1987. Translated from the German and edited by L. Boron, C. Christenson and B. Smith. MR 920366
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
D. Cordero-Erausquin, M. Fradelizi, and B. Maurey, The (B) conjecture for the Gaussian measure of dilates of symmetric convex sets and related problems, J. Funct. Anal. 214 (2004), no. 2, 410–427. MR 2083308, DOI 10.1016/j.jfa.2003.12.001
S. Dancs and B. Uhrin, On a class of integral inequalities and their measure-theoretic consequences, J. Math. Anal. Appl. 74 (1980), no. 2, 388–400. MR 572660, DOI 10.1016/0022-247X(80)90136-5
Matthieu Fradelizi and Arnaud Marsiglietti, On the analogue of the concavity of entropy power in the Brunn-Minkowski theory, Adv. in Appl. Math. 57 (2014), 1–20. MR 3206519, DOI 10.1016/j.aam.2014.02.004
R. J. Gardner, The Brunn-Minkowski inequality, Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 3, 355–405. MR 1898210, DOI 10.1090/S0273-0979-02-00941-2
Richard J. Gardner and Artem Zvavitch, Gaussian Brunn-Minkowski inequalities, Trans. Amer. Math. Soc. 362 (2010), no. 10, 5333–5353. MR 2657682, DOI 10.1090/S0002-9947-2010-04891-3
G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge, at the University Press, 1952. 2d ed. MR 0046395
R. Henstock and A. M. Macbeath, On the measure of sum-sets. I. The theorems of Brunn, Minkowski, and Lusternik, Proc. London Math. Soc. (3) 3 (1953), 182–194. MR 56669, DOI 10.1112/plms/s3-3.1.182
María A. Hernández Cifre and Jesús Yepes Nicolás, Refinements of the Brunn-Minkowski inequality, J. Convex Anal. 21 (2014), no. 3, 727–743. MR 3243816
Daniel Hug, Günter Last, and Wolfgang Weil, A local Steiner-type formula for general closed sets and applications, Math. Z. 246 (2004), no. 1-2, 237–272. MR 2031455, DOI 10.1007/s00209-003-0597-9
J. Kampf, The parallel volume at large distances, Geom. Dedicata 160 (2012), 47–70. MR 2970042, DOI 10.1007/s10711-011-9669-x
R. Latała, On some inequalities for Gaussian measures, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 813–822. MR 1957087
L. Leindler, On a certain converse of Hölder’s inequality. II, Acta Sci. Math. (Szeged) 33 (1972), no. 3-4, 217–223. MR 2199372
A. Livne Bar-on, The (B) conjecture for uniform measures in the plane, Geometric Aspects of Functional Analysis, Lecture Notes in Mathematics Volume 2116, 2014, 341-353.
A. Marsiglietti, Concavity properties of extensions of the parallel volume, Mathematika, Available on CJO2015 doi:10.1112/S0025579314000369.
Piotr Nayar and Tomasz Tkocz, A note on a Brunn-Minkowski inequality for the Gaussian measure, Proc. Amer. Math. Soc. 141 (2013), no. 11, 4027–4030. MR 3091793, DOI 10.1090/S0002-9939-2013-11609-6
András Prékopa, Logarithmic concave measures with application to stochastic programming, Acta Sci. Math. (Szeged) 32 (1971), 301–316. MR 315079
András Prékopa, On logarithmic concave measures and functions, Acta Sci. Math. (Szeged) 34 (1973), 335–343. MR 404557
C. Saroglou, Remarks on the conjectured log-Brunn-Minkowski inequality, To appear in Geom. Dedicata.
Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1993. MR 1216521, DOI 10.1017/CBO9780511526282
J. Steiner, Über parallele Flächen, Monatsbericht der Akademie der Wissenschaften zu Berlin (1840), pp. 114–118.