Abstract
Let K and L be two convex bodies in \({{\mathbb {R}}^5}\). Assume that their orthogonal projections K|H and L|H onto every 4-dimensional subspace H are directly SU(2)-congruent, i.e., they coincide up to a SU(2)-rotation for some complex structure in H and a translation in H. We prove that the bodies coincide up to a translation and a reflection in the origin, provided that the set of diameters of one of the bodies is contained in a finite union of two-dimensional subspaces of \({{\mathbb {R}}^5}\). We obtain this result as a consequence of a more general statement about a functional equation on the unit sphere.
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The author is supported in part by U.S. National Science Foundation Grant DMS-1600753.
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Ryabogin, D. On a functional equation related to convex bodies with SU(2)-congruent projections. Geom Dedicata 190, 151–156 (2017). https://doi.org/10.1007/s10711-017-0234-0
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DOI: https://doi.org/10.1007/s10711-017-0234-0