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Journal of Convex Analysis 21 (2014), No. 2, 415--424
Copyright Heldermann Verlag 2014

Mean-Value Inequalities for Convex Functions and the Chebysev-Vietoris Inequality

Pal Fischer
Dept. of Mathematics and Statistics, University of Guelph, Guelph, Ontario N1G 2W1, Canada

Zbigniew Slodkowski
Dept. of Mathematics, Statistics and Computer Science, University of Illinois, Chicago, IL 60607-7045, U.S.A.


\def\R{\mathbb{R}} It is shown that if $B=[-b_1, b_1] \times \cdots \times [-b_n,b_n] \subset \R^n,$ where $b_i>0$ for $i=1,...,n\,,$ and if $A$ is a convex and compact subset of $B$ of positive Lebesgue measure, which is preserved by reflections with respect to all coordinate hyperplanes $x_i=0$ for $i=1,...,n \,,$ then $A$ is convexly majorized by $B,$ i.e., for every continuous convex function $v$ defined over $B,$ the mean of $v$ over $A$ is not exceeding the mean of $v$ over $B.$ In the proof an n-dimensional extension of the integral form of the Chebysev inequality, which was given by L. Vietoris [{\it Eine Verallgemeinerung eines Satzes von Tschebyscheff}, Univ. Beograd Publ. Elektrotehn, Fak. Ser. Mat. Fiz 461-497 (1974) 115-117], is used.

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