Abstract
This paper is to provide some new generalizations of the Pick Theorem. We first derive a point-set version of the Pick Theorem for an arbitrary bounded lattice polyhedron. Then, we use the idea of a weight function of [2] to obtain a weighted version. Other Pick type theorems known to the author for the integral lattice Z 2 are reduced to some special cases of this generalization. Finally, using an idea of Ehrhart [6] and the Pick Theorem, we give a direct proof of the reciprocity law for Dedekind sums. The ideas and methods presented here may be pushed to higher dimensions.
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