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

P. Fischer ; Z. Slodkowski

Journal of convex analysis, Tome 21 (2014) no. 2, pp. 415-424.

Résumé

\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.