Steinhaus conditions for convex polyhedra

Abstract

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.