**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.

**Preprint version available:** pdf

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