Characterization of semicontinuity of solutions to abstract optimization problems

Abstract

We develop a theory of best simultaneous approximations for closed downward sets in a conditionally complete lattice Banach space X with a strong unit. We study best simultaneous approximation in X by elements of downward and normal sets, and give necessary and sufficient conditions for any element of best simultaneous approximation by a closed subset of X. We prove that a downward subset of X is strictly downward if and only if each its boundary point is simultaneous Chebyshev.