Journal of Geometry

, 110:16 | Cite as

Discrete volumes of lattice polyhedra via vector analysis

  • Yasuzo NishimuraEmail author


Pick’s theorem relates the number of lattice points to the area for a lattice polygon. Diaz and Robins gave a proof of Pick’s theorem by using the Weierstrass \(\wp \)-function and complex analysis. As an analogue to lattice convex polyhedra, Reeve’s theorem is known as a solid version of Pick’s theorem. In this paper, we study counting lattice points on a lattice polyhedron by using vector analysis, and we extend Reeve’s theorem to nonconvex polyhedral complexes.


Lattice polyhedron Pick’s theorem Gauss divergence theorem 

Mathematics Subject Classification

Primary 52B20 52B10 



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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Faculty of Education, Humanities and Social SciencesUniversity of FukuiFukuiJapan

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