Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-24T02:25:54.349Z Has data issue: false hasContentIssue false
  • English
  • Français

Two problems on convex bodies

Published online by Cambridge University Press:  24 October 2008

H. T. Croft
H. T. Croft
Affiliation:
PeterhouseCambridge
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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.