Hollow polytopes of large width
Giulia Codenotti, Francisco Santos · Proceedings of the American Mathematical Society · 2019
We construct the first known hollow lattice polytopes of width larger than dimension: a hollow lattice polytope (resp., a hollow lattice simplex) of dimension 14 14 (resp., 404 404 ) and of width 15 15 (resp., 408 408 ). We also construct a hollow (nonlattice) tetrahedron of width 2 + 2 2+\sqrt 2 , and we conjecture that this is the maximum width among 3 3 -dimensional hollow convex bodies. We show that the maximum lattice width grows (at least) additively with d d . In particular, the constructions above imply the existence of hollow lattice polytopes (resp., hollow simplices) of arbitrarily large dimension d d and width ≃ 1.14 d \simeq 1.14 d (resp., ≃ 1.01 d \simeq 1.01 d ).