On the continual Rubik's cube
The main result of the paper is Theorem 1. Let \( f \) and \( g \) be two continuous functions on the unit sphere \(S^{n-1} \subset {\mathbb R}^n \), \( n \geq 3\), and let their restrictions onto any one-dimensional great circle \(E \subset S^{n-1}\) coincide after some rotation \(\phi_E\) of this circle: \(f(\phi_E(\theta))=g(\theta) \) for all \(\theta \in E\). Then \(f(\theta)=g(\theta)\) or \(f(\theta)=g(-\theta)\) for all \(\theta \in S^{n-1}\). This answers the question formulated in [\textit{V. P. Golubyatnikov}, Sib. Math. J. 36, No. 2, 265--269 (1995); translation from Sib. Mat. Zh. 36, No. 2, 301--307 (1995; Zbl 0865.52004)]. Using this theorem, the author obtains the following. Theorem 2. Let \( n \geq 3\) and let \(K\) and \(L\) be two convex bodies in \( {\mathbb R}^n \) containing the origin in their interior. Then \(K=L\) or \(K= - L\), provided the projections \( K \mid H \), \(L \mid H\) onto any two-dimensional subspace \(H\) of \({\mathbb R}^n\) are rotations of each other around the origin. Theorem 3. Let \(K\) and \(L\) be two star-shaped bodies with respect to the origin in \( {\mathbb R}^n\), \( n \geq 3\). Then \(K=L\) or \(K= - L\), provided the sections \( K \cap H \), \(L \cap H\) by any two-dimensional subspace \(H\) of \(\, {\mathbb R}^n\) are rotations of each other around the origin. These two theorems are closely related to higher-dimensional open problems formulated in [\textit{R. J. Gardner}, Geometric tomography. 2nd ed. Cambridge: Cambridge University Press (2006; Zbl 1102.52002)] for more general cases \( 2 \leq \dim H \leq n-1\).
scientific article; zbMATH DE number 6108934
scientific article