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Diameters in Typical Convex Bodies

Published online by Cambridge University Press:  20 November 2018

Imre Bárány  and
Tudor Zamfirescu
Imre Bárány
Affiliation:
Academy of Sciences, Budapest, Hungary
Tudor Zamfirescu
Affiliation:
University of Dortmund, Dortmund, Germany
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The most usual diameters in the world are those of a sphere and they all contain its centre. More generally, a chord of a convex body in Rd is called a diameter if there are two parallel supporting hyperplanes at the two endpoints of the chord.

It is easily seen that there are points on at least two diameters. From a result of Kosiński [6] proved in a more general setting it follows that every convex body has a point lying on at least three diameters. Does a typical convex body behave like a sphere and contain a point on infinitely or even uncountably many diameters?

But what is a typical convex body? The space 𝒦 of all convex bodies (d-dimensional compact convex sets) in Rd, equipped with the Hausdorff metric, is a Baire space.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 42 , Issue 1 , 01 February 1990 , pp. 50 - 61
Copyright
Copyright © Canadian Mathematical Society 1990

References

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