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We construct a hollow lattice polytope (resp. a hollow lattice simplex) ofdimension $14$ (resp.$~404$) and of width $15$ (resp.$~408$). They are thefirst known hollow lattice polytopes of width larger than dimension. We alsoconstruct a hollow (non-lattice) tetrahedron of width $2+\sqrt2$ and conjecturethat 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 latticepolytopes (resp. hollow simplices) of arbitrarily large dimension $d$ and width$\simeq 1.14 d$ (resp.$~\simeq 1.01 d$).

Hollow polytopes of large width