# Basic Notions and Definitions of Multidimensional Integer Geometry

• Oleg Karpenkov
Chapter
Part of the Algorithms and Computation in Mathematics book series (AACIM, volume 26)

## Abstract

In this chapter we generalize integer two-dimensional notions and definitions of Chap. . As in the planar case, our approach is based on the study of integer invariants. Further, we use them to study the properties of multidimensional continued fractions. First, we introduce integer volumes of polytopes, integer distances, and integer angles. Then we express volumes of polytopes, integer distances, and integer angles in terms of integer volumes of simplices. Finally, we show how to compute integer volumes of simplices via certain Plücker coordinates in the Grassmann algebra (this formula is extremely useful for the computation of multidimensional integer invariants of integer objects contained in integer planes). We conclude this chapter with a discussion of the Ehrhart polynomials, which one can consider a multidimensional generalization of Pick’s formula in the plane.

## Keywords

Integer Geometry Integer Intensity Ehrhart Polynomial Integer Angle Integer Invariants
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## References

1. 18.
M. Beck, S. Robins, Computing the Continuous Discretely: Integer-Point Enumeration in Polyhedra. Undergraduate Texts in Mathematics (Springer, New York, 2007) Google Scholar
2. 54.
E. Ehrhart, Sur les polyèdres rationnels homothétiques à n dimensions. C. R. Math. Acad. Sci. Paris 254, 616–618 (1962)
3. 88.
J.-M. Kantor, K.S. Sarkaria, On primitive subdivisions of an elementary tetrahedron. Pac. J. Math. 211(1), 123–155 (2003)
4. 115.
S.V. Konyagin, K.A. Sevastyanov, Estimate of the number of vertices of a convex integral polyhedron in terms of its volume. Funkc. Anal. Prilozh. 18(1), 13–15 (1984)