δ-convexity in normed linear spaces

For some given positive δ, a function f :D (contained in or equal) X → R is called δ-convex if it satisfies the Jensen inequality f(xλ) ≤(1-λ)f(x0)+λf(x1) for all x0,x1 (is element of) D and xλ:=(1-λ)x0+λx1 (is element of) [x0, x1] satisfying norm(x0-x1) ≥ δ, norm(xλ-x0) ≥ δ/2 and norm(xλ-x1)≥δ/2. In this paper, we introduce δ-convex sets and show that a function f:D (contained in or equal) X → R is δ-convex if the level set {x (is element of) D:f(x)+ξ(x)≤α} is δ-convex for every continuous linear functional ξ (is element of) X* and for every real α. Some optimization properties such as constant property on affine sets, and analytical properties such as boundedness on bounded sets, local boundedness, conservation and infection of δ-convex functions are presented. (author)