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Shadow-boundaries and cuts of convex polytopes

Published online by Cambridge University Press:  26 February 2010

Peter Kleinschmidt  and
Udo Pachner
Peter Kleinschmidt
Affiliation:
Institut för Mathematik, Ruhr-Universität Bochum, Bochum, West Germany
Udo Pachner
Affiliation:
Institut för Mathematik, Ruhr-Universität Bochum, Bochum, West Germany
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Let P be a (convex) d-polytope in the Euclidean space Ed and p a point of Ed not contained in P or in a supporting hyperplane of a facet of P (we use the terminology of Grunbaum [2]). The part of the boundary of P which is “visible” from p, i.e. the union of those facets whose supporting hyperplanes separate p and P, form a (d – l)-ball, whose boundary is a (d – 2)-sphere S (all balls, spheres and manifolds to be considered are piecewise-linear). The boundary-complex ℬ(P) induces a subdivision of S, which we call a (sharp) shadow-boundary ofP. is combinatorially isomorphic to the boundary complex of a poly tope which is a central projection of P from the point p on a hyperplane.

Type
Research Article
Information
Mathematika , Volume 27 , Issue 1 , June 1980 , pp. 58 - 63
Copyright
Copyright © University College London 1980

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