 
Summary: Classifying Hyperplanes in Hypercubes
Oswin Aichholzer Franz Aurenhammer y
Institute for Theoretical Computer Science
Graz University of Technology
Klosterwiesgasse 32/2
A8010 Graz, Austria
email: oaich@igi.tugraz.ac.at
auren@igi.tugraz.ac.at
Abstract
We consider hyperplanes spanned by vertices of the unit dcube. We classify these
hyperplanes by parallelism to coordinate axes, by symmetry of the dcube vertices
they avoid, as well as by socalled hullhonesty. (Hullhonest hyperplanes are those
whose intersection gure with the dcube coincides with the convex hull of the dcube
vertices they contain; they do not cut dcube edges properly.) We describe relation
ships between these classes, and give the exact number of hullhonest hyperplanes,
in general dimensions. An experimental enumeration of all spanned hyperplanes up
to dimension eight showed us the intrinsic diÆculty of developing a general enu
meration scheme. Motivation for considering such hyperplanes stems from coding
theory, from linear programming, and from the theory of machine learning.
Research supported by the Austrian Ministry of Science and the Jubilaumsfond der
