If a line goes through the boundary of a convex set, does that line intersect with exactly two boundary points of the convex region?

Asked Mar 03 '15 at 04:37

Active Mar 03 '15 at 04:37

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I have found the following theorem that is often cited from the text Convex Figures by Yaglom and Boltyanskii:

A bounded figure in $\mathbb{R}^2$ is convex iff every straight line passing through an arbitrary interior point of the figure intersects the boundary of the figure in exactly two points.

I cannot find the text, nor a proof of this statement, but it seems like it would generalize to higher dimensions. Is this true?

asked Mar 03 '15 at 04:37

Eager Student

2

You can find one part of the reference here. By the way, using these questions, one can assume that the figure is closed. – Watson Nov 06 '16 at 20:16
The implication $\implies$ should remain true in higher dimensions. – Watson Nov 06 '16 at 20:23
@Watson what is the reference there? – J. Moeller Feb 12 '18 at 03:13
1

@Prototank : if you mean the first link I gave, it's from Convex Figures by I.M. Yaglom and V.G. Boltyanskii. – Watson Feb 12 '18 at 07:19

If a line goes through the boundary of a convex set, does that line intersect with exactly two boundary points of the convex region?

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