# Thread: Proving that a shape S is a sphere.

1. ## Proving that a shape S is a sphere.

Let $S\subset {\mathbb{R}}^3$ with the following properties:

1. For any line $l$:

$|l\cap S|=2$ or $|l\cap S|=1$ or $|l\cap S|=0$

2. For any plane $P$:

$|P\cap S|=4$ or $|P\cap S|=1$ or $|P\cap S|=0$

Prove that $S$ is a sphere.

2. ## Re: Proving that a shape S is a sphere.

What does the | | sign mean? First I thought it must be the number of elements of the set. But in that case |PS| = infinity, not 4.