Giulia Codenotti: Lattice reduced polytopes and the flatness question

Abstract: We call a convex body, or a polytope, lattice reduced if it does not strictly contain any other convex body of the same lattice width--a parameter measuring "how flat" the body is with respect to the lattice of integer points. We show structural properties of lattice reduced convex bodies: that they are polytopes, with not too many vertices; and then investigate the connection to the flatness problem: determining, for each dimension, the maximum lattice width of convex bodies without interior integer points. We will then delve into further directions of study and open questions.