Outer Normal Transforms of Convex Polytopes

Abstract

For a d-dimensional convex polytope $P\subset {\mathbb {R}}^d$, the outer normal unit vectors of its facets span another d-dimensional polytope, which is called the outer normal transform of P and denoted by $P^*$. It seems that this transform, which clearly differs from the usual polarity transform, was not seriously investigated until now. We derive results about polytopes having natural properties with respect to this transform, like self-duality or equality for the composition of it. Among other things we prove that if P is a convex polytope inscribed in the unit sphere with the property that the circumradii of its facets are all equal, then $(P^*)^*$ coincides with P. We also prove that the converse is true when P or $P^*$ has at most 2d vertices. For the three-dimensional case, we characterize the family of those tetrahedra T such that T and $T^*$ are congruent.