  Generalized permutahedra are a class of polytopes with many interesting
combinatorial subclasses. We introduce pruned inside-out polytopes, a
generalization of inside-out polytopes introduced by Beck--Zaslavsky (2006,
arXiv:math/0309330), which have many applications such as recovering the famous
reciprocity result for graph colorings by Stanley. We study the integer point
count of pruned inside-out polytopes by applying classical Ehrhart polynomials
and Ehrhart--Macdonald reciprocity. This yields a geometric perspective on and
a generalization of a combinatorial reciprocity theorem for generalized
permutahedra by Aguiar--Ardila (arXiv:1709.07504) and Billera--Jia--Reiner
(2009, arXiv:math/0606646). Applying this reciprocity theorem to hypergraphic
polytopes allows us to give an arguably simpler proof of a recent combinatorial
reciprocity theorem for hypergraph colorings by Aval--Karaboghossian--Tanasa
(2020, arXiv:1806.08546). Our proof relies, aside from the reciprocity for
generalized permutahedra, only on elementary geometric and combinatorial
properties of hypergraphs and their associated polytopes.
