  In this work we blend interpolation theory with numerical integration,
constructing an interpolator based on integrals over $n$-dimensional balls. We
show that, under hypotheses on the radius of the $n$-balls, the problem can be
treated as an interpolation problem both on a collection of $(n-1)$-spheres $
S^{n-1} $ and multivariate point sets, for which a wide literature is
available. With the aim of exact quadrature and cubature formulae, we offer a
neat strategy for the exact computation of the Vandermonde matrix of the
problem and propose a meaningful Lebesgue constant. Problematic situations are
evidenced and a charming aspect is enlightened: the majority of the theoretical
results only deal with the centre of the domains of integration and are not
really sensitive to their radius. We flank our theoretical results by a large
amount of comprehensive numerical examples.
