Weak Chebyshev spaces on locally ordered topology space and the related continuous metric selections (Q1096110)

Let X be a locally connected and compact Hausdorff space and G an n- dimensional Z-subspace of C(X), the space of continuous real functions with sup-norm. \(P_ G(f)\) is the set of nearest functions in G to f, and \(P_ G\) has a continuous selection iff there is a continuous mapping S: C(X)\(\to G\) with \(S(f)\in P_ G(f)\). The following statements are equivalent: (1) \(P_ G\) has a unique pointwise-Lipschitz-continuous semi-additive selection; (2) \(P_ G\) has a continuous selection; (3) Every nonzero g in G has at most n zeros and at most n-1 zeros with sign changes; (4) G is a weak Chebyshev space and every nonzero g in G has at most n zeros; (5) G is semi-definite and every f in C(X) has a unique alternation element; (6) G is semi-definite and every nonzero g in G has at most n zeros.