  On a convex surface $S$, the antipodal map $F$ associates to a point $p$ the
set of farthest points from $p$, with respect to the intrinsic metric. $S$ is
called a Steinhaus surface if $F$ is a single-valued involution. We prove that
any convex polyhedron has an open and dense set of points $p$ admitting a
unique antipode $F_p$, which in turn admits a unique antipode $F_{F_p}$,
distinct from $p$. In particular, no convex polyhedron is Steinhaus.
