Suppose a compact convex body PsubsetBbbR3 has only polygonal orthogonal projections onto a plane. Does this imply that P is a convex polytope?
This question occurred to me when I was making exercises for my book. I figured this is probably easy and well known, but the literature hasn't been any help. One remark: if the number of sides of all polygons is bounded by n, the problem might be easier. Furthermore, if P is assumed to be a convex polytope, this elegant paper by Chazelle-Edelsbrunner-Guibas (1989) gives a (perhaps, unexpectedly large) sharp expO(nlogn) upper bound on the number of vertices of P (ht Csaba Toth who generalized this to higher dimensions).
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