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Metrics for sets of convex bodies

Published online by Cambridge University Press:  26 February 2010

G. C. Shephard  and
R. J. Webster
G. C. Shephard
Affiliation:
Department of Pure Mathematics, The University, Birmingham, 15
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Extract

In many investigations into the properties of convex bodies, authors have made use of distance functions ρ(K1K2) which give a measure of the “nearness” of two convex bodies K1 and K2. Sometimes they have introduced new functions to deal with particular problems. The purpose of this paper is to compare and contrast the properties of four of these functions, namely all those (so far as we are aware) which occur in the literature and have the property that they are metrics on the set of all convex bodies of some given dimension.

Type
Research Article
Information
Mathematika , Volume 12 , Issue 1 , June 1965 , pp. 73 - 88
Copyright
Copyright © University College London 1965

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References

1. Blaschke, W., Kreis und Kugel (Leipzig, 1916).CrossRefGoogle Scholar
2. Blumenthal, L. M., Distance Geometry (Oxford, 1953).Google Scholar
3. Bonnesen, T. and Fenchel, W., Theorie der Konvexen Körper (Berlin, 1934; Reprint, New York, 1948).Google Scholar
4. Dinghas, A., “Ueber das Verhalten der Entfernung zweier Punktmengen bei gleichzeitig Symmetrisierung derselben”, Arch. Moth., 8 (1957), 46–51.CrossRefGoogle Scholar
5. Eggleston, H. G., Convexity (Cambridge, 1958).CrossRefGoogle Scholar
6. Eggleston, H. G., Problems in Euclidean space: Applications of convexity (New York and London, 1957).Google Scholar
7. Gladysz, S., Marczewski, E. and Ryll-Nardzewski, C., “Concerning distances of sets and distances of functions”, Colloquium Mathematicum, 8 (1961), 71–75.CrossRefGoogle Scholar
8. Halmos, P. R., Measure theory (Princeton, 1950).Google Scholar
9. Hausdorff, F., Set theory (New York, 1949).Google Scholar
10. John, F., “Extrernum problems with inequalities as subsidiary conditions”, Courant Anniversary Volume, 1948, 187–204.Google Scholar
11. Marczewski, E. and Steinhaus, H., “On a certain distance of sets and the corresponding distance of functions”, Colloquium Mathematicum, 6 (1958), 319–327.Google Scholar
12. Menger, K., “Untersuchungen über allgemeine Metrik”, Math. Annalen, 100 (1928), 75–163.CrossRefGoogle Scholar
13. Shephard, G. C., “Inequalities between mixed volumes of convex sets”, Mathematika, 7 (1960), 125–138.CrossRefGoogle Scholar
14. Walsh, J. L., “On the transformation of convex point sets”, Ann. of Math. (2), 22 (1921), 262–266.CrossRefGoogle Scholar