On Lattice Diameter Segments and A Discrete Borsuk Partition Problem

The lattice diameter of a bounded set $S \subset \mathbb{R}^d$ measures the maximal number of lattice points in a segment whose endpoints are lattice points in $S$. Such a segment is called a lattice diameter segment of $S$. This simple invariant yields interesting applications and challenges. We describe a polynomial-time algorithm that computes lattice diameter segments of lattice polygons and show that computing lattice diameters of semi-algebraic sets in dimensions three and higher is NP-hard. We prove that the function that counts lattice diameter segments in dilations of a lattice polygon is eventually a quasi-polynomial in the dilation factor. We also study the number of directions that lattice diameter segments can have. Finally, we prove a Borsuk-type theorem on the number of parts needed to partition a set of lattice points such that each part has strictly smaller lattice diameter.