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Classifying Hyperplanes in Hypercubes (Extended Abstract)

Summary: Classifying Hyperplanes in Hypercubes
(Extended Abstract)
Oswin Aichholzer 1 Franz Aurenhammer
Institute for Theoretical Computer Science
Graz University of Technology
Klosterwiesgasse 32/2
A­8010 Graz, Austria
e­mail: oaich@igi.tu­graz.ac.at
1 Introduction
Among the simplest high­dimensional geometric objects is the d­dimensional hypercube
(d­cube) C d = [0; 1] d . Dispite of its simple definition, C d has been an object of study
from various different points of view. The theory of convex polytopes provides classical
results concerning sections and projections of hypercubes; see Coxeter [3] and Gr¨unbaum
[6]. Purely combinatorial properties of C d , mainly involving certain subgraphs formed
by its edges and vertices (the latter are just the various d­tuples of binary digits) have
been investigated extensively in coding theory and in communication theory; see, e.g.,
[4, 2, 11, 5]. Many easily stated questions concerning the geometry of C d are still unsettled.
A long­standing elementary conjecture on hypercube space fillings (Keller's conjecture) has
been recently disproved [7].


Source: Aurenhammer, Franz - Institute for Theoretical Computer Science, Technische Universität Graz
Technische Universität Graz, Institute for Software Technology


Collections: Computer Technologies and Information Sciences