Lattice point inequalities for centered convex bodies (Q2808167)

Let \(K\) be a convex body in the Euclidean space \(\mathbb R^d\). Define NEWLINE\[NEWLINEG(K) = \left| K \cap \mathbb Z^d \right|,NEWLINE\]NEWLINE in other words \(G(K)\) counts the number of integer lattice points in \(K\). Finding bounds on the function \(G(K)\) under various conditions on \(K\) has been an important research direction in discrete and convex geometry even prior to the classical work of Minkowski. There is a variety of results in the case when \(K\) is \(\mathbf 0\)-symmetric; in particular, \textit{U. Betke} et al. [Discrete Comput. Geom. 9, No. 2, 165--175 (1993; Zbl 0771.52007)] established the bound NEWLINE\[NEWLINEG(K) \leq \left \lfloor{ \frac{2}{\lambda_1(K)} + 1}\right \rfloor^d,NEWLINE\]NEWLINE where \(\lambda_1(K)\) is the first successive minimum of the \(\mathbf 0\)-symmetric convex body \(K\).NEWLINENEWLINEThe interesting paper under review is mostly concerned with the more general case when the convex body \(K\) is not necessarily \(\mathbf 0\)-symmetric. Specifically, under the assumption that \(K\) has its centroid at \(\mathbf 0\), the authors prove that NEWLINE\[NEWLINEG(K)< 2^d \left( \frac{2}{\lambda_1(K)} + 1 \right)^d.NEWLINE\]NEWLINE One implication of this result is that if \(\mathbf 0\) is the only integer lattice point in the interior of \(K\), then \(G(K)< 6^d\). The authors further conjecture a stronger upper bound on \(G(K)\) under the same conditions, specifically NEWLINE\[NEWLINEG(K) \leq \binom{d + \lceil{ \lambda_1(K)^{-1} (d+1) \rceil}}{d}.NEWLINE\]NEWLINE They prove this conjecture for the case when \(K\) is a \(d\)-simplex and discuss the case of equality in their inequality. They also demonstrate that their conjecture implies the well-known Ehrhart conjecture, asserting that \(d\)-simplex has the smallest volume out of all convex bodies \(K\) with centroid at \(\mathbf 0\), which is the only interior integer lattice point of \(K\). Finally, the authors also prove their conjecture for planar convex bodies \(K\) with centroid at \(\mathbf 0\), which is the only interior integer lattice point of \(K\), establishing in that case the sharp bound \(G(K) \leq 10\). The proofs employ a variety of nice geometric techniques, related to Ehrhart theory.