Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-23T14:57:45.027Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Gruber, P., “Zur Charakteisierung konvexer Körper: Über einen Satz von Rogers und Shephard”, Math. Ann., 181 (1969), 189–200.CrossRefGoogle Scholar
2.Gruber, P., “Zur Charakterisierung konvexer Körper: Uuml;ber einen Satz von Rogers und Shephard. II”, Math. Ann. (to appear).Google Scholar
3.Grünbaum, B., Convex polytopes (John Wiley and Sons: London, New York and Sydney, 1967).Google Scholar
4.Rogers, C. A. and Shephard, G. C., “The difference body of a convex body”, Arch. Math., 8 (1957), 220–233.CrossRefGoogle Scholar
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