A FAMILY of inequalities concerning inner products of vectors and functions began with Cauchy. The extensions and generalizations later led to the inequalities of Schwarz, Minkowski and Holder. The well known Holder inequality involves the inner product of vectors measured by Minkowski norms. In this paper, another step of extension is taken so that a Holder type inequality may apply to general...

A precise description of the convexity of Gaussian measures is provided by sharp Brunn-Minkowski type inequalities due to Ehrhard and Borell. We show that these are manifestations of a game-theoretic mechanism: a minimax variational principle for Brownian motion. As an application, we obtain a Gaussian improvement of Barthe’s reverse Brascamp-Lieb inequality.

In this paper, we establish a Farkas-Minkowski type alternative theorem under the supposition of nearly semiconvexlike set-valued maps. Based on the alternative theorem and some other lemmas, we present necessary optimality conditions and sufficient optimality conditions for set-valued vector optimization problems with extended inequality constraints in a sense of weak E-minimizers.

This paper develops a significant extension of E. Lutwak’s dual Brunn-Minkowski theory, originally applicable only to star-shaped sets, to the class of bounded Borel sets. The focus is on expressions and inequalities involving chord-power integrals, random simplex integrals, and dual affine quermassintegrals. New inequalities obtained include those of isoperimetric and Brunn-Minkowski type. A n...

in this paper, two pairs of new inequalities are given, which decompose two hilbert-type inequalities.