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. 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 $\exp O(n \log n)$ upper bound on the number of vertices of $P$ (ht Csaba Toth who generalized this to higher dimensions).

reallynot a polytope. Now what? $\endgroup$ – Igor Pak Feb 18 '10 at 4:18