Bounds for the lattice point enumerator (Q1179137)

The paper gives some upper and lower bounds of the lattice point enumerator \(G(K,\mathbb{L})=\hbox{card}(K\cap\mathbb{L})\). Here \(K\) is a convex body in Euclidean \(d\)-space \(E^ d\), \(d\geq 2\), and \(\mathbb{L}\) is a lattice in \(E^ d\) with \(\hbox{det} \mathbb{L}>0\). The bounds are given in terms of the intrinsic volumes of \(K\), and in terms of the numbers \(D_ i(\mathbb{L})=\min\{|\hbox{det} \mathbb{L}_ i|\): all \(i\)-dimensional sublattices \(\mathbb{L}_ i\subset\mathbb{L}\}\) for \(i=1,\dots,d\) and \(D_ 0(\mathbb{L})=1\). One of the bounds is \(G(K,\mathbb{L})\leq\sum^ d_{i=0}[i!V_ i(K)/D_ i(\mathbb{L})]\). For \(d=2\) it is improved to the sharp bound \(G(K,\mathbb{L})\leq\sum^ 2_{i=0}[V_ i(K)/D_ i(\mathbb{L})]\).