Homogeneity Sets for Jensen-Convex Functions

Abstract

For a convex subset ∆ of a real vector space X and a function f: ∆ → ℝ, the homogeneity set H_f is defined by

$$ {H_{f}}: = \left\{ {\lambda \in \left[ {0,1} \right]:f\left( {\lambda x + \left( {1 - \lambda } \right)y} \right) \le \lambda f\left( x \right) + \left( {1 - \lambda } \right)f\left( y \right){\kern 1pt} for{\kern 1pt} all{\kern 1pt} x,y{\kern 1pt} \in {\kern 1pt} \Delta } \right\} $$

. In this paper, homogeneity sets of Jensen-convex functions are discussed.