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Thread: Proving that a shape S is a sphere.

  1. #1
    MHF Contributor Also sprach Zarathustra's Avatar
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    Proving that a shape S is a sphere.

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

    1. For any line $\displaystyle l$:

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

    2. For any plane$\displaystyle P$:

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

    Prove that $\displaystyle S$ is a sphere.
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  2. #2
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    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.
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