δ-convexity in normed linear spaces

For some given positive delta, a function f : D subset of or equal to X --> IR is called delta-convex if it satisfies the Jensen inequality f(x(lambda)) less than or equal to (1 - lambda)f(x(0)) + lambdaf(x(1)) for all x(0), x(1), is an element of D and x(lambda) := (1 - lambda)x(0) + lambdax(1) is an element of [x(0), x(1)] satisfying parallel tox(0) - x(1)parallel to greater than or equal to delta, parallel tox(lambda) - x(0)parallel to greater than or equal to delta/2 and parallel tox(lambda) - x(1) parallel to greater than or equal to delta/2 [Hu, T. C., Klee, V., Larman, D. (1989). Optimization of globally convex functions. SIAM J. Control Optim. 27:1026-1047]. In this paper, we introduce delta-convex sets and show that a function f : D subset of or equal to X --> IR is delta-convex iff the level set {x is an element of D : f(x) + xi(x) less than or equal to alpha} is delta-convex for every continuous linear functional xi is an element of X* and for every real a. 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 delta-convex functions are presented.