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Is the following true:

Let, $S$ is a compact connected $(n-1)$-dimensional surface in $\mathbb{R}^n$ s.t., for every point $p \in S$ there is a neighborhood $\mathcal{V}_p \subset \mathbb{R}^n$ of $p$ s.t., $\Sigma \cap S$ is a circle (equivalent to $\mathbb{S}^1$) for every $2$-dim plane $\Sigma$ s.t., $(\Sigma \cap S) \subset \mathcal{V}_p$. Then, $S$ is a sphere.

If instead of local planes near points on the surface we had every $2$-dim planar section are circles then it is obviously a sphere by a maximum diameter argument.

Equivalently, we can rephrase it as $\Sigma \cap S$ being equivalent to $\mathbb{S}^{n-2}$ for every $(n-1)$-dim planar section $\Sigma \cap S \subset \mathcal{V}_p$ for all $p \in S$.

But how can we pass from this local information to $S$ being a sphere?

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    $\begingroup$ Does not Dupin indicatrix solve it? That is, take a section by a plane spanned by two principal directions of curvature... $\endgroup$ Nov 20, 2018 at 18:39
  • $\begingroup$ @IvanIzmestiev Thank you very much! $\endgroup$
    – sciona
    Nov 20, 2018 at 18:53

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