Best approximation by downward sets with applications
H. Mohebi, Alexander M. Rubinov · Analysis in Theory and Applications · 2006
We develop a theory of downward sets for a class of normed ordered spaces. We study best approximation in a normed ordered space X by elements of downward sets, and give necessary and sufficient conditions for any element of best approximation by a closed downward subset of X. We also characterize strictly downward subsets of X, and prove that a downward subset of X is strictly downward if and only if each its boundary point is Chebyshev. The results obtained are used for examination of some Chebyshev pairs (W, x), where x ε X and W is a closed downward subset of X.