Hollow polytopes of large width

Abstract

We construct a hollow lattice polytope (resp. a hollow lattice simplex) of dimension $14$ (resp.$404$) and of width $15$ (resp.$408$). They are the first known hollow lattice polytopes of width larger than dimension. We also construct a hollow (non-lattice) tetrahedron of width $2+\sqrt{2}$ and conjecture that this is the maximum width among $3$-dimensional hollow convex bodies. We show that the maximum lattice width grows (at least) additively with $d$. In particular, the constructions above imply the existence of hollow lattice polytopes (resp. hollow simplices) of arbitrarily large dimension $d$ and width $\simeq 1.14d$ (resp.$\simeq 1.01d$).

17 pages, 3 figures