Related to the questions [mathoverflow.net question No. 137850](https://mathoverflow.net/questions/137850/A-question-about-a-question-about-3-dimensional-convex-bodies) and [mathoverflow.net question No. 39127](https://mathoverflow.net/questions/39127/is-the-sphere-the-only-surface-all-of-whose-projections-are-circles-or-can-we), is there a 3-dimensional convex body other than a ball whose perpendicular projections in all directions are of the same area?



(A question about)${}^3$ 3-dimensional convex bodies
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.


