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arXiv · 2407.12310

A characterization of translated convex bodies

Efren Morales-Amaya
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Abstract

In this work we present a theorem regarding two convex bodies $K_1, K_2\subset \mathbb{R}^{n}$, $n\geq 3$, and two families of sections of them, given by two families of tangent planes of two spheres $S_i\subset \textrm{int}\textrm{ } K_i$, $i=1,2$ such that, for every pair $\Pi_1$, $\Pi_2$ of parallel supporting planes of $S_1$, $S_2$, respectively, which are corresponding (this means, that the outer normal vectors of the supporting half spaces determined by the two planes have the same direction), the sections $\Pi_1\cap K_1$, $\Pi_2\cap K_2$ are translated, the theorem claims that if $S_1$, $S_2$ have the same radius, the bodies are translated, otherwise, the bodies are also spheres.

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Efren Morales-Amaya. 2024-07-17. A characterization of translated convex bodies. https://arxiv.org/abs/2407.12310

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