Suppose a compact convex body $P \subset \Bbb R^3$ 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](http://www.math.ucla.edu/~pak/book.htm). 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](http://www.cs.princeton.edu/~chazelle/pubs/ComplexityCuttingComplexes.pdf) elegant paper by Chazelle-Edelsbrunner-Guibas (1989) gives a (perhaps, unexpectedly large) sharp $\exp O(n \log n)$ upper bound on the number of vertices of $P$ (ht Csaba Toth who [generalized](http://math.ucalgary.ca/~cdtoth/cx-stabbing-rev.pdf) this to higher dimensions). 



Do plane projections determine a convex polytope?
Theorem 4.1 of [this paper by Klee](http://www.ams.org/mathscinet-getitem?mr=105651) says yes. Moreover, the result generalizes to higher dimensions for projections of arbitrary dimension $\ge 2$.


