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Summary: Classifying Hyperplanes in Hypercubes
Oswin Aichholzer Franz Aurenhammer y
Institute for Theoretical Computer Science
Graz University of Technology
Klosterwiesgasse 32/2
A-8010 Graz, Austria
e-mail: oaich@igi.tu-graz.ac.at
auren@igi.tu-graz.ac.at
Abstract
We consider hyperplanes spanned by vertices of the unit d-cube. We classify these
hyperplanes by parallelism to coordinate axes, by symmetry of the d-cube vertices
they avoid, as well as by so-called hull-honesty. (Hull-honest hyperplanes are those
whose intersection gure with the d-cube coincides with the convex hull of the d-cube
vertices they contain; they do not cut d-cube edges properly.) We describe relation-
ships between these classes, and give the exact number of hull-honest 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
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