Abstract: 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 (resp., 404) and of width 15 (resp., 408). We also construct a hollow (nonlattice) tetrahedron of width 2 + ?2, and we 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  1.14d (resp.,  1.01d).

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