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In this talk, we will discuss two related topics in discrete geometry.

 

First, we study slices of convex polytopes by affine hyperplanes, focusing on the sequence of possible numbers of vertices in such slices. We discuss combinatorial and geometric properties of these vertex sequences, with particular emphasis on the case of the cyclic polytopes and hypercubes.

 

Second, we study the lattice diameter of a lattice polygon: the intersection of the polytope with a line containing the maximum number of lattice points. We present a polynomial-time algorithm to compute all lattice diameters in dimension 2, and we show that the problem becomes NP-hard in higher dimensions. We also investigate the possible directions that these diameters can take. 

 

This is joint work with Jesús A. De Loera, Gyivan López, and Anouk Brose.