Summary: Classifying Hyperplanes in Hypercubes
Oswin Aichholzer 1 Franz Aurenhammer
Institute for Theoretical Computer Science
Graz University of Technology
A8010 Graz, Austria
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  and Gr¨unbaum
. 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 .