Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-28T10:58:48.117Z Has data issue: false hasContentIssue false
  • English
  • Français

Vector sums of Valentine convex sets

Published online by Cambridge University Press:  24 October 2008

H. G. Eggleston
H. G. Eggleston
Affiliation:
Royal Holloway College
Get access Rights & Permissions [Opens in a new window]

Summary

A set X in Euclidean space is Valentine n–convex, or simply n–convex' if it has the following property. If X contains a subset Y consisting of n distinct points then X also contains the points of at least one segment with end points in Y. We show here that the vector sum of two plane compact 3-convex sets is 5-convex (which complements the result of I. D. Calvert(1) that the intersection of two plane compact 3-convex sets is 5-convex) and that the vector sum of a plane connected compact 3-convex set with itself is 4-convex. These results are not true in 4 dimensional space. It is an open question whether or not they are true in 3-dimensional space.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 92 , Issue 1 , July 1982 , pp. 17 - 19
Copyright
Copyright © Cambridge Philosophical Society 1982

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

REFERENCES

(1)Calvert, D. I. Generalisations of Convexity. Thesis, London 1979.Google Scholar
(2)Eggleston, H. G.Math. Proc. Cambridge Phil. Soc. 77 (1975), 525–528.CrossRefGoogle Scholar