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Suppose we have a 3-D shape $S$ with a center $C$, so that a point $p$ is in $S$ if and only if for any direction $\vec d$, $p$ is contained within a cylinder of radius $1$, extending infinitely both ways in the direction $\vec d$ and with its axis passing through $C$.

Is the resulting shape a sphere of radius $1$?

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I expect that the answer is "yes", but I don't know how to prove it. – Joe Z. Mar 15 '13 at 18:35
up vote 3 down vote accepted

Assuming that you’ve seen that the sphere of radius $1$ is contained in every such cylinder, I’ll give you this hint: consider a point $P$ that's not in the unit sphere. Can’t you find one of your cylinders that similarly doesn’t contain $P$?

To focus your mind, you might coordinatize lightly, putting your center $C$ at the origin of Cartesian $3$-space. And permit yourself to choose your bad point with very good coordinates.

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Yeah, it was simple. – Joe Z. Mar 15 '13 at 18:42
(This wasn't homework, by the way. I'm not even in class for a while; the question just popped up.) – Joe Z. Mar 15 '13 at 18:43
It’s a natural question to wonder about. – Lubin Mar 15 '13 at 18:44

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