On conditionally δ-convex functions

Abstract

Let X be a real vector space, V a subset of X and δ ≧ 0 a given number. We say that f: V → ℝ is a conditionally δ-convex function if for each convex combination t ₁ υ ₁ + … + t _n υ _n of elements of V such that t ₁ υ ₁ + … + t _n υ _n ∈ V the following inequality holds true:

$$ f(t_1 v_1 + \cdots + t_n v_n ) \leqq t_1 f(v_1 ) + \cdots + t_n f(v_n ) + \delta . $$

We prove that f: V → ℝ is conditionally δ-convex if and only if there exists a convex function $\tilde f$: conv V → [−∞, ∞) such that

$$ \tilde f(v) \leqq f(v) \leqq \tilde f(v) + \delta for v \in V. $$

In case X = ℝⁿ some conditions equivalent to conditional δ-convexity are also presented.