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Characterization of certain polytopes by intersection properties of their translates

Published online by Cambridge University Press:  26 February 2010

Rolf Schneider
Rolf Schneider
Affiliation:
Ruhr-Universität Bochum, Germany.
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Extract

The object of the present note is to prove the following theorem. †

Theorem. Let P be a convex d-polytope in d-dimensional real vector space Rd (d ≥ 2). If each d-polytope which is the intersection of two translates of P has the same number of vertices as P, then P is a direct linear sum ‡ of simplices.

Type
Research Article
Information
Mathematika , Volume 16 , Issue 2 , December 1969 , pp. 276 - 282
Copyright
Copyright © University College London 1969

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References

1.Gruber, P., “Zur Charakteisierung konvexer Körper: Über einen Satz von Rogers und Shephard”, Math. Ann., 181 (1969), 189–200.CrossRefGoogle Scholar
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3.Grünbaum, B., Convex polytopes (John Wiley and Sons: London, New York and Sydney, 1967).Google Scholar
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5.Schneider, R., “Über die Durchschnitte translationsgleicher konvexer Korper und eine Klasse konvexer Polyeder”, Abh. Math. Sent. Univ. Hamburg, 30 (1967), 118–128.CrossRefGoogle Scholar