Classification of three-dimensional multistoried completely hollow convex marked pyramids.

*(English. Russian original)*Zbl 1167.11311
Russ. Math. Surv. 60, No. 1, 165-166 (2005); translation from Usp. Mat. Nauk 60, No. 1, 169-170 (2005).

In this note the author gives a result on the solution of a problem in the geometry of integer lattices and its application to the theory of multidimensional continued fractions. Consider an integer two-dimensional plane and an integer point in the complement of this plane. Let the Euclidean distance from the given point to the given plane be \(l\). Let \(l_0\) denote the minimal non-zero Euclidean distance to the given plane from the integer points of the three-dimensional plane spanned by the two-dimensional plane and the integer point under consideration. The ratio \(l/l_0\) is called the integer distance from the given integer point to the given integer plane. An integer pyramid is called multistorey if the integer distance from the vertex of this pyramid to the plane of its base is greater than one.

Theorem 1. Any multistorey completely empty convex three-dimensional marked pyramid is integer-affinely equivalent to exactly one of the marked pyramids in the list ‘M-W’ (given in the paper).

Note that up to now the only known compact two-dimensional faces (of sails of multidimensional continued fractions) with planes at an integer distance from the origin greater than one were either triangular or quadrangular. Here the author gives a complete integer-linear classification of these faces.

Theorem 1. Any multistorey completely empty convex three-dimensional marked pyramid is integer-affinely equivalent to exactly one of the marked pyramids in the list ‘M-W’ (given in the paper).

Note that up to now the only known compact two-dimensional faces (of sails of multidimensional continued fractions) with planes at an integer distance from the origin greater than one were either triangular or quadrangular. Here the author gives a complete integer-linear classification of these faces.

Reviewer: Olaf Ninnemann (Berlin)