On Slater like inequality for vectors transformed by a doubly stochastic matrix with control function

Abstract

In this work, we deal with a Slater type inequality
designed for a symmetric convex function
and for a collection of vectors
transformed by a doubly stochastic matrix.
In doing so, we use an additional convex control function.
In the case when the composition of the control function and of the underlying convex function is Schur-concave,  such an approach leads to
a refinement of the standard Slater inequality. 
Special cases are also considered.