Geodesic
Integrals of logarithmically concave functions
Matematičeskie zametki, Tome 20 (1976) no. 6, pp. 843-845 
V. A. Tomilenko. Integrals of logarithmically concave functions. Matematičeskie zametki, Tome 20 (1976) no. 6, pp. 843-845. http://geodesic.mathdoc.fr/item/MZM_1976_20_6_a5/
@article{MZM_1976_20_6_a5,
     author = {V. A. Tomilenko},
     title = {Integrals of logarithmically concave functions},
     journal = {Matemati\v{c}eskie zametki},
     pages = {843--845},
     year = {1976},
     volume = {20},
     number = {6},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1976_20_6_a5/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

In this note we consider integrals of the form $$ \int_Af(x,y)\,dy\stackrel{def}=I(x,A), $$ where $f$ is a finite logarithmically concave function in $E^{n+m}$ and $A$ is a convex subset of the space $E^m$. For any pair of convex sets $A$ and $B$ and any $x_1,x_2\in E^n$ we establish the inequality $$ I(\lambda x_1+(1-\lambda)x_2,\lambda A+(1-\lambda)B)\ge I^\lambda(x_1,A)I^{1-\lambda}(x_2,B)\quad0<\lambda<1. $$