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Characterizations of the sphere by means of visual cones: an alternative proof of Matsuura's theorem Amaya, Efren Morales


In this work we prove that if there exists a point $p\in \mathbb{R}^n$ and a smooth convex body $M$ in $\mathbb{R}^n$, $n\geq 3$, contained in the interior of the unit ball of $\mathbb{R}^n$, such that $M$ looks centrally symmetric, and $p$ appears as the centre, from each point of $\mathbb{S}^{n-1}$, then $M$ is an sphere. Using this result we derived, straightaway, a well known characterization of the sphere due to S. Matsuura

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