Abstract

It is shown that if B=[−b1,b1]×⋯×[−bn,bn]⊂Rn,B=[-b_1, b_1] \times \cdots \times [-b_n,b_n] \subset \R^n, where bi>0b_i>0 for i=1,...,n ,i=1,...,n\,, and if AA is a convex and compact subset of BB of positive Lebesgue measure, which is preserved by reflections with respect to all coordinate hyperplanes xi=0x_i=0 for i=1,...,n ,i=1,...,n \,, then AA is convexly majorized by B,B, i.e., for every continuous convex function vv defined over B,B, the mean of vv over AA is not exceeding the mean of vv over B.B. In the proof an n-dimensional extension of the integral form of the Chebysev inequality, which was given by L. Vietoris [Eine Verallgemeinerung eines Satzes von Tschebyscheff, Univ. Beograd Publ. Elektrotehn, Fak. Ser. Mat. Fiz 461-497 (1974) 115-117], is used

Author information

Contact details are reproduced from the original publication and may be historical.

Pal Fischer

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

pfischer@uoguelph.ca

Zbigniew Slodkowski

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

zbigniew@uic.edu

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Suggested citation

P. Fischer, Z. Slodkowski. “Mean-Value Inequalities for Convex Functions and the Chebysev-Vietoris Inequality.” Journal of Convex Analysis 21 (2014), No. 2, 415–424. https://doi.org/10.68381/jca21021

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Published by Heldermann Verlag, 2014. Rights now held by Banach Press LLC.