Minimal pairs of polytopes and their number of vertices

Abstract

We define what is called Blaschke difference for polytopes as an inverse operation to Blaschke addition. Using this operation we give a new algorithm to reduce and find a minimal pair of polytopes from the given class of the Rådström-Hörmander lattice containing a pair of polytopes in IR². This method gives a better algorithmic insight and easy to handle than the one given by Handschug (1989). We also prove that a pair of polytopes in the plane is minimal if and only if the sum of the number of their vertices is minimal in the class. However, it is shown in the paper that, this last statement does not hold true in general for higher dimensional spaces.