KAIST Discrete Math Seminar

We study the convex hull of the symmetric moment curve U_k(t)=(cost, sint, cos3t, sin3t, …., cos(2k-1)t, sin(2k-1)t) in R^2k and provide deterministic constructions of centrally symmetric polytopes with a record high number faces. In particular, we prove the local neighborliness of the symmetric moment curve, meaning that as long as k distinct points t₁, …, t_k lie in an arc of a certain length φ_k > π/2, the points U_t1, …, U_tk span a face of the convex hull of U_k(t). In this talk, I will use the local neighborliness of the symmetric moment curve to construct d-dimensional centrally symmetric 2-neighborly polytopes with approximately 3^d/2 vertices. This is joint work with Alexander Barvinok and Isabella Novik.