Abstract
Let X be a list of vectors that is unimodular and let B X be the box spline defined by X. We discuss the proof of the following conjecture by Holtz and Ron: every real-valued function on the set of interior lattice points of the zonotope defined by X can be extended to a function on the whole zonotope of the form p(D)B X in a unique way, where p(D) is a differential operator that is contained in the so-called internal \(\mathcal{P}\)-space. We construct an explicit solution to this interpolation problem in terms of truncations of the Todd operator. As a corollary we obtain a slight generalisation of the Khovanskii-Pukhlikov formula that relates the volume and the number of lattice points in a smooth lattice polytope.
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The author was supported by a Junior Research Fellowship of Merton College (University of Oxford).
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Lenz, M. (2015). On a Conjecture of Holtz and Ron Concerning Interpolation, Box Splines, and Zonotopes. In: Benedetti, B., Delucchi, E., Moci, L. (eds) Combinatorial Methods in Topology and Algebra. Springer INdAM Series, vol 12. Springer, Cham. https://doi.org/10.1007/978-3-319-20155-9_15
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DOI: https://doi.org/10.1007/978-3-319-20155-9_15
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