Abstract: We obtain some characterizations of almost interpolation configurations of points with respect to finite-dimensional functional spaces. Particularly, a Schoenberg-Whitney type characterization which is valid for any multivariate spline space relative to an arbitrary partition of a domain $A\subset\RR^m$ is presented. As a closely related problem we investigate sectional structure of finite-dimensional spaces of real functions on a topological space $A$. It is shown that under some reasonable restrictions on $A$ any space of this sort may be considered as piecewise almost Chebyshev.
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