Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-13T10:17:25.522Z Has data issue: false hasContentIssue false
  • English
  • Français

Twenty Problems on Convex Polyhedra Part II

Published online by Cambridge University Press:  03 November 2016

G. C. Shephard
G. C. Shephard*
Affiliation:
School of Mathematics and Physics, University of East Anglia, Wilberforce Road, Norwich, NOR 88 C
Get access Rights & Permissions [Opens in a new window]

Extract

Let P be a given convex polyhedron and L any plane. Then if L∩P is not empty it is called a plane section of P. Every plane section of P is either a convex polygon, an edge or a vertex of P.

VI. If a convex polyhedron P has v vertices (v ≥4), does there always exist a plane section of P which is a convex n-gon with n ≥ √ v ?

A more general statement of this problem is to ask whether there exists a constant d (0 < d < 1) such that every convex polyhedron P with v vertices possesses a plane section which is an n-gon with n ≥ vd. Problem VI is then equivalent to asking whether d ≤ ½

Type
Research Article
Information
The Mathematical Gazette , Volume 52 , Issue 382 , December 1968 , pp. 359 - 367
Copyright
Copyright © Mathematical Association 1968

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. Coxeter, H. S. M., Regular Polytopes (Methuen: London and New York), 1948 (Second edition 1963).Google Scholar
2. Grünbaum, B., Convex Polytopes (Wiley: London and New York), 1967.Google Scholar
3. Lyustemik, L. A., Convex Figures and Polyhedra (Russian). English translation by Smith, T. Jefferson (Dover: New York), 1963.Google Scholar
3a. Lyustemik, L. A., English translation by Barnett, D. L. (Heath: Boston), 1966.Google Scholar
4. Rouse-Ball, W. W., Mathematical Recreations and Essays (Macmillan: London), 1892; (Eleventh edition 1939).Google Scholar
5. Sommerville, D. M. Y., An Introduction to the Geometry of N Dimensions (Methuen: London), 1929; (Dover: New York), 1958.Google Scholar