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On unique reconstruction of convex and visible compacta by their projections. (Russian) Zbl 0738.52004

This is a contribution to the question: What conditions imply that if two compacta have congruent projections on 2-dimensional flats, then the compacta are congruent? Here are two typical results of the paper.
1. Suppose \(W_ 1\), \(W_ 2\) are simply connected compacta in \(\mathbb{R}^ n\) with the property that through each \(n-2-\ell\) dimensional plane which does not intersect \(W_ i\) there can be drawn an \(n-2\) dimensional plane which does not intersect \(W_ i\); here \(0<\ell\leq n-2\). If the projections of \(W_ 1\), \(W_ 2\) on any 2-dimensional plane are properly congruent and have no rotational symmetry in that plane, then \(W_ 1\), \(W_ 2\) can be made to coincide by a translation of central symmetry. 2. If \(V_ 1\), \(V_ 2\) are compact convex sets in \(\mathbb{R}^ 3\) whose projections on any plane are properly similar and have no rotational symmetry, then \(V_ 1\), \(V_ 2\) can be made to coincide by a homothety.

MSC:

52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
52A30 Variants of convex sets (star-shaped, (\(m, n\))-convex, etc.)
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