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Characterizations of Spherical Neighbourhoods

Published online by Cambridge University Press:  20 November 2018

C. M. Petty  and
J. M. Crotty
C. M. Petty
Affiliation:
University of Missouri, Columbia, Missouri
J. M. Crotty
Affiliation:
University of Missouri, Columbia, Missouri
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If Σ is a specified class of metric spaces and M ∈ Σ, then the characterization problem is to find necessary and sufficient conditions which distinguish the spherical neighbourhoods (open spheres) of M among a specified class of subsets of M.

In a metric space M the notation pqr means p ≠ q ≠ r and pq + qr = pr.M is said to be uniformly locally externally convex if there exists δ > 0 such that if p, q ∈ M, p ≠ q, and pq < δ, then there exists r ∈ M such that the relation pqr subsists. We will prove the following result.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 2 , 01 April 1970 , pp. 431 - 435
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Blaschke, W., Rothe, H., and R., Weitzenböck, Doppelspeichenkurven, Aufgabe 552, Arch. Math. Phys. 27 (1918), 82.Google Scholar
2. Blumenthal, L. M., Theory and applications of distance geometry (Oxford Univ. Press, London, 1953).Google Scholar
3. Busemann, H., The geometry of geodesies (Academic Press, New York, 1955).Google Scholar
4. Dirac, G. A., Ovals with equichordal points, J. London Math. Soc. 27 (1952), 429–437.Google Scholar
5. Dulmage, L., Tangents to ovals with two equichordal points, Trans. Roy. Soc. Canada Sect. III (3) 48 (1954), 7–10.Google Scholar
6. Hsiang, W., Another characterization of circles, Amer. Math. Monthly 69 (1962), 142–143.Google Scholar
7. Klee, V., Can a plane convex body have two equichordal points'?, Amer. Math. Monthly 76 (1969), 54–55.Google Scholar
8. Naïmark, M. A., Normed rings (P. Noordhoff, Gröningen, 1964).Google Scholar
9. Siiss, W., Eibereiche mit ausgezeichneten Punkten; Sehnen-, Inhalts- und Umfangspunkte, Töhoku Math. J. 25 (1925), 86–98.Google Scholar
10. Wirsing, E., Zur Analytizitdt von Doppelspeichenkurven, Arch. Math. 9 (1958), 300–307.Google Scholar
11. Whittlesey, E. F., Fixed points and antipodal points, Amer. Math. Monthly 70 (1963), 807–821.Google Scholar