ABSTRACT
Define where B d is the unit ball of R d . P r is clearly a lattice polytope. It follows from (more general) results of Andrews, Konyagin and Sevastyanov, Schmidt (cf. [1], [8]), and [13]) that the number k-dimensional faces, f k (P r ) of P r satisfies
Here, and in what follows, for functions f, g the notation means that f=Cg for some constant C depending only on the dimension. In addition, f˜g means that
It is proved in Bárány and Larman [4] that the above inequality is sharp, apart from the
implied constant:
Further, it is shown in [4] (see Remark 1 on page 173) that the number of lattice points on the boundary of P r is r
vertices of P r . More formally, we have
Theorem 1. Actually, we will prove the following slightly stronger statement. Call a facet of P r
rich if it contains a lattice point in its relative interior. Theorem 2. The number of rich facets of P r is rd
For d≥3 we first prove Theorem 2, which readily implies Theorem 1. The order is reversed for d=2: we give a direct proof of Theorem 1. In addition, we show that Theorem 1 yields Theorem 2 in the planar case.
