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Intersections of tangent convex curves

Part of: General convexity

Published online by Cambridge University Press:  09 April 2009

Tudor Zamfirescu
Tudor Zamfirescu
Affiliation:
Abteilung Mathematik, Universität Dortmund, D-4600 Dortmund 50, Postfach 500 500, Federal Republic of Germany
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Abstract

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We prove in this note that if two convex curves are internally tangent (lie on the same side of the common tangent), then in most cases they also cut each other infinitely many times. It follows that, in a certain sense, if two convex curves have an odd number of common points, then in general they are externally tangent (lie on different sides of the common tangent). The sense of the expressions most cases and in general remains to be precised.

MSC classification

Secondary: 52A10: Convex sets in $2$ dimensions (including convex curves)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 31 , Issue 4 , December 1981 , pp. 456 - 458
Copyright
Copyright © Australian Mathematical Society 1981

References

[1]Gruber, P. M., ‘Die meisten konvexen Körper sind glatt, aber nicht zu glatt’, Math. Ann. 229 (1977), 259–266.CrossRefGoogle Scholar
[2]Klee, V., ‘Some new results on smoothness and rotundity in normed linear spaces’, Math. Ann. 139 (1959), 51–63.CrossRefGoogle Scholar
[3]Zamfirescu, T., ‘The curvature of most convex surfaces vanishes almost everywhere’, Math. Z. 174 (1980), 135–139.CrossRefGoogle Scholar