Parameter spaces of separating hyperplanes

Abstract

Every pair of relatively disjoint polytopes is dual to the parameter space of all their separating hyperplanes, which is also a polytope. For a polytope whose interior is disjoint from the relative interior of another polytope, the parameter space of all separating hyperplanes is a polytope of the same dimension. One face of this parameter space parametrizes the separating hyperplanes that also simultaneously support both polytopes. A separating hyperplane corresponds to a vertex of this face if and only if no other hyperplanes support the polytopes at the same intersection points. If all the vertices of the polytopes have all their coordinates in an ordered field, then the same results and their proofs hold with the same ordered field.

Keywords:

• convex sets
• minimal separation
• rigid support
• spaces of hyperplanes

Classification:

• Primary: 52B12
• 52A40
• 90C05
• Secondary: 52A20
• 52B55
• 52C35

Acknowledgments

This work was supported in part by a Professional Leave from Eastern Washington University. I thank the referees for suggestions that greatly improved the manuscript.