Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-26T01:10:08.446Z Has data issue: false hasContentIssue false
  • English
  • Français

In which metric spaces are parallel bodies of closed sets closed?

Published online by Cambridge University Press:  20 January 2009

Gerald Beer
Gerald Beer
Affiliation:
California State University, Los Angeles, Los Angeles, California 90032
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let X be a metric space with metric d and for each x in X let Bλ[x] denote the closed ball of radius λ about x. Following Valentine [15] if K⊂X and λ is positive, then we call the set Bλ[K]=∪x∈KBλ[x] the λ-parallel body of K. The following fact is obvious.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 26 , Issue 2 , June 1983 , pp. 265 - 273
Copyright
Copyright © Edinburgh Mathematical Society 1983

References

REFERENCES

1.Beer, G., Starshaped sets and the Hausdorff metric, Pacific J. Math. 61 (1975), 21–27.CrossRefGoogle Scholar
2.Beer, G., The approximation of upper semicontinuous multifunctions by step multifunctions, Pacific J. Math. 87 (1980), 11–19.CrossRefGoogle Scholar
3.Beer, G., Upper semicontinuous functions and the Stone approximation theorem, J. Approximation Theory 34 (1982), 1–11.CrossRefGoogle Scholar
4.Berge, C., Topological spaces (Oliver and Boyd, Edinburgh, 1963).Google Scholar
5.Borges, C., A study of multivalued functions, Pacific J. Math 23 (1967), 451–461.CrossRefGoogle Scholar
6.Brown, M., Sets of constant distance from a planar set, Michigan Math. J. 19 (1972), 321–323.CrossRefGoogle Scholar
7.Dolecki, S. and Rolewicz, S., Metric characterizations of upper semicontinuity, J. Math. Anal. Appl. 69 (1979), 146–152.CrossRefGoogle Scholar
8.Ferry, S., When ɛ-boundaries are manifolds, Fund. Math. 90 (1976), 199–210.CrossRefGoogle Scholar
9.Gariepy, R. and Pepe, W., On the level sets of a distance function in a Minkowski space, Proc. Amer. Math. Soc. 31 (1972), 255–259.CrossRefGoogle Scholar
10.Gindifer, M., On generalized spheres, Fund. Math. 38 (1951), 167–178.CrossRefGoogle Scholar
11.Goudini, G., On set-values mappings, Rev. Roum. Math. Pures Appl. 22 (1) (1977), 53–67.Google Scholar
12.Hindman, N., Basically bounded sets and a generalised Heine-Borel theorem, Amer. Math. Monthly 80 (1973), 549–552.CrossRefGoogle Scholar
13.Hu, S., Introduction to general topology (Holden Day, San Francisco, 1966).Google Scholar
14.Nadler, S., Hyperspaces of sets (Marcel Dekker, New York, 1978).Google Scholar
15.Valentine, F., Convex sets, (McGraw-Hill, New York, 1964).Google Scholar