  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 $2+\sqrt2$ 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
$\simeq 1.14 d$ (resp.$~\simeq 1.01 d$).
