δ-Convexity in Normed Linear Spaces

Abstract

For some given positive δ, a function f : D ⊆ X → ℝ is called δ-convex if it satisfies the Jensen inequality f(x _λ) := (1 − λ)f(x ₀) + λf(x ₁) for all x ₀, x ₁ ∈ D and x _λ ≔ (1 − λ)x ₀ + λx ₁ ∈ [x ₀, x ₁] satisfying ‖x ₀ − x ₁‖ ≥ δ, ‖x _λ − x ₀‖ ≥ δ/2 and ‖x _λ − x ₁‖ ≥ δ/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 δ-convex sets and show that a function f : D ⊆ X → ℝ is δ-convex iff the level set {x ∈ D : f(x) + ξ(x) ≤ α} is δ-convex for every continuous linear functional ξ ∈ 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.

Key Words:

• Generalized convexity
• Rough convexity
• δ-convexity
• Minimum
• Maximum
• Local boundedness
• Boundedness

AMS Classification 1991:

• 26B25
• 26A51
• 52A01

Acknowledgments

The authors gratefully acknowledge many helpful suggestions of Professor Dr. Sc. Hoang Xuan Phu during the preparation of the paper. They wish to thank the Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste, Italy, where this paper was written, for the invitation and hospitality. Financial supports of the Swedish International Development Cooperation Agency and Mathematics Research Fellowship at ICTP are acknowledged.