Generalizations of Dobrushin's Inequalities and Applications

Abstract

Letf:Rn→Rbe a seminorm and let (ei)1≤i≤nbe the canonical base ofRn. DenoteM=12 maxr,sf(er−es),K=maxrf(er). We prove the inequality f(x)\\le M\\biggl(\\sum_{i=1}^n|x_i|\\biggr)+(K-M)\\biggl|\\sum_{i=1}^n x_i\\biggr|,\\qquad x=(x_1,x_2,\\ldots,x_n)\\in{\\Bbb R}^n. We use the above inequality to prove some generalizations of Dobrushin's inequalities and a generalization of an inequality due to J. E. Cohenet al.(Linear Algebra Appl.179, 1993, 211–235). Hilbert space generalizations of the above inequalities are proved using Levi's reduction theorem. As special cases of our results we obtain several inequalities given previously by Adamovici, Djokovic, Hlawka, and Hornich.