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Counterexamples related to rotations of shadows of convex bodies. (English) Zbl 1366.52002

Summary: We construct examples of two convex bodies \(K,L\) in \(\mathbb{R}^n\), such that every projection of \(K\) onto an \((n-1)\)-dimensional subspace can be rotated to be contained in the corresponding projection of \(L\), but \(K\) itself cannot be rotated to be contained in \(L\). We also find necessary conditions on \(K,L\subset \mathbb{R}^3\) to ensure that \(K\) can be rotated to be contained in \(L\) if all the two-dimensional projections have this property.

MSC:

52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
52A38 Length, area, volume and convex sets (aspects of convex geometry)
44A12 Radon transform
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