Well-Posed Uniform Solvability of Convex Optimization Problems on a Uniform Differentiable Closed Convex Set

In this paper, we first give the definition of uniformly differentiable set and give the definitions of sets P(A, eta, r) and P-A,P-delta(f). Secondly, we prove that if the set A is bounded closed convex set, then A is uniformly differentiable if and only if for any epsilon, eta, r > 0, there exists delta = delta(epsilon, eta,r) > 0 such that & Vert;x-y & Vert; < epsilon whenever f is an element of P(A,eta,r), y is an element of P-A,P-delta (f) and x is an element of P-A(f). Moreover, we also prove that if A is abounded closed convex set in a finite-dimensional space X, then A is differentiable if and only if A is uniformly differentiable. Finally, we give some examples of uniformlydifferentiable set. Therefore, we extend some conclusions (SIAM J. Optim. Vol. 30, No. 1, pp. 490-512).