# Surfaces of constant projected area

Generalizing the well-known variety of plane curves of constant width, I'm wondering about three-dimensional surfaces of constant projected area.

Question: If $A$ is a (bounded) subset of $\mathbb R^3$, homeomorphic to a closed ball, such that the orthogonal projection of $A$ onto a plane has the same area for all planes, is $A$ necessarily a sphere? If not, what are some other possibilities?

Wikipedia mentions a concept of convex shapes with constant width, but that's different.

(Inspired by the discussion about spherical cows in comments to this answer -- my question is seeking to understand whether there are other shapes of cows that would work just as well).

-
This is explicitly not what you want to know, but there's such a beautiful thread on MO about shapes with spherical projections mathoverflow.net/questions/39127 that it would be a pity not to mention it. –  t.b. Aug 7 '12 at 17:11

Thank you! ${}$ –  Henning Makholm Aug 12 '12 at 20:19