On the boundary of an extremal body (Q1567452)

If \(L\) is a lattice in euclidean space \(E^n\), then, as shown by \textit{H. Minkowski} [see Geometrie der Zahlen, Leipzig-Berlin (1910; JFM 41.0239.03)], a 0-symmetric convex body \(K\) which contains no points of \(L\) other that 0 in its interior has volume at most \(2^nv(D)\), with \(v(D)\) the volume of the basic parallepiped of \(L\). If \(v(K)= 2^nv(D)\), then \(K\) is called extremal; in this case, \(K\) is a polytope. Minkowski also established further restrictions on how many points of \(L\) can lie in the boundary of \(K\), or in various faces of \(K\). In this paper, the author produces refinements of such results, and of those of Venkov, the reviewer and earlier ones of the author.