On the number of lattice points in convex symmetric bodies and their duals (Q1182843)

Let \(K\subset\mathbb{R}^ n\) be a convex, centrally symmetric, bounded, absorbing set, \(vol(K)\) its standard volume, \(K^*\) its polar convex set, and \(\# (K\cap\mathbb{Z}^ n)\) the number of lattice points in \(K\). Authors prove that \(\# (K\cap\mathbb{Z}^ n)/(\# (K^*\cap\mathbb{Z}^ n)\text{vol}(K))\) is bounded below and above by positive constants depending on \(n\) but not on \(K\), and they give various applications of this result.