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Two problems on convex bodies

Published online by Cambridge University Press:  24 October 2008

H. T. Croft
H. T. Croft
Affiliation:
PeterhouseCambridge
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Extract

We solve two problems on convex bodies stated on p. 38 of S. M. Ulam's book, A collection of mathematical problems (New York, 1960).

Problem 1. This problem is due to Mazur. In three-dimensional Euclidean space there is given a convex surface W and a point O in its interior. Consider the set V of all points P defined by the requirement that the length of the interval OP is equal to the area of the plane section of W through O and perpendicular to OP. Is the (centrally symmetric) set V a convex surface?

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 58 , Issue 1 , January 1962 , pp. 1 - 7
Copyright
Copyright © Cambridge Philosophical Society 1962

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References

† See, for example, Eggleston, H. G., Convexity (Cambridge, 1958), p. 32, Ex. 1·11. Such a point is called a regular point.CrossRefGoogle Scholar

† As will be seen, here and below in Theorem 5, the conclusion is true if the second derivative is uniformly bounded; the proof is very similar to that given.