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).

These are called bodies of constant brightness. A convex body that has both constant width and constant brightness is a Euclidean ball. But non-spherical convex bodies of constant brightness do exist; the first was found by Blaschke in 1916. See: Google and related MSE thread.
