Theorem shows that parallel sections by tangent planes imply a solid sphere in a convex body.
We prove the following theorem concerning a convex body in the space with a sphere in its interior: If every two parallel cross-sections of the body by planes tangent to the sphere are translates of each other by the vector connecting the tangency points, then the body is a solid sphere concentric with the given sphere.
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Aguilar-Arteaga et al. (2025) studied this question.