On the Boundary of an Extremal Body

Abstract: In this paper we shall investigate the boundary of an extremal body $K$. Using the characterization of the extremal bodies proved by Venkov and McMullen (Theorem 1) we give two theorems (Theorems 4, 5) determining the relative position of the lattice vectors on the boundary of $K$. These statements are analogues of Theorem 2 and Theorem 3 proved for Dirichlet-Voronoi cells in [H]. In the third paragraph we investigate the connection between the simplicity of a face and the property that it contains lattice points in its relative interior (Theorems 6, 7, 8).

\item{[H]} Horvath, A. G.: On the DIRICHLET-VORONOI cell of unimodular lattices. Geometriae Dedicata 63 (1996), 183-191.

Full text of the article:

• Compressed DVI file (23510 Bytes)
• Compressed Postscript file (76912 Bytes)