TITLE: (A question about)${}^3$ 3-dimensional convex bodies
QUESTION [15 upvotes]: Related to the questions mathoverflow.net question No. 137850 and mathoverflow.net question No. 39127, is there a 3-dimensional convex body other than a ball whose perpendicular projections in all directions are of the same area?

REPLY [15 votes]: A convex body $K\subset\mathbb{R}^n$ all of whose $(n-1)$-dimensional projections have the same $n-1$-content is known as a body of constant brightness, by analogy with bodies of constant width. The theory is very similar to that of bodies of constant width. The surface area measure $dS_K(\mathbf{x})$ takes the place of the support function $h_K$. The brightness in the direction $\mathbf{u}$ is given by $V(K_\mathbf{u}) = \tfrac{1}{2}\int |\langle\mathbf{x},\mathbf{u}\rangle| dS_K(\mathbf{x})$. If we expand $S_K$ in spherical harmonics $\sum_{l,m}s_{l,m}Y_{l,m}$, we get that $V(K_\mathbf{u}) = \sum_{l,m} c_l s_{l,m} Y_{l,m}(\mathbf{u})$, and $c_l=0$ whenever $l$ is odd. Therefore, $K$ has constant brightness if and only if $\frac{S_K(U)+S_K(-U)}{|U|}$ is constant over all Borel sets $U\subset S^{n-1}$ (that is, apart from a constant term, $S_K$ is antisymmetric). From the existence theorem for the Minkowski problem, we can easily construct examples.