Abstract
In this chapter, we study various structural connectedness-type properties of approximating sets (among which we consider Chebyshev sets, suns, moons, uniqueness sets, and so on). By structural characteristics of sets one usually understands properties of linearity, finite-dimensionality, convexity, connectedness of various kinds, and smoothness of sets. From results of such kind one may derive necessary and sufficient conditions for a set to have certain important approximative properties such as solarity and lunarity. We shall give direct theorems of geometric approximation theory in which approximative properties of sets are derived from their structural characteristics and put forward converse theorems in which from approximative characteristics of sets one derives their structural properties.
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