Abstract
In this paper we consider a class of Schur-concave functions with some measure properties. The isoperimetric inequality and Brunn-Minkowsky’s inequality for such kind of functions are presented. Applications in geometric programming and optimization theory are also derived.
Highlights
About years ago, the properties concerning such notions as length, area, volume as well as the probability of events were abstracted under the banner of the word measure
We study some measure properties of a special class of Schur-concave functions which will be revealed via some discrete versions of isoperimetric inequality and Brunn-Minkowsky’s inequality
We present a discrete version of isoperimetric inequality related to a special class of Schur-concave functions
Summary
About years ago, the properties concerning such notions as length, area, volume as well as the probability of events were abstracted under the banner of the word measure. In [ ] a class of analytic inequalities for Schur-convex functions that are made of solutions of a second order nonlinear differential equation was established. These analytic inequalities are used to infer some geometric inequalities such as isoperimetric inequality. We present a discrete version of isoperimetric inequality related to a special class of Schur-concave functions. Let us consider a more difficult problem concerning the classical isoperimetric inequality for convex bodies from RN. Notice that the well-known measures - perimeter, area, volume - are Schur-concave functions. We are able to present the discrete form of the isoperimetric inequality in the context of this type of n-dimensional volume functions. We have the following isoperimetric inequality: Fn(x , . . . , xn)
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