Nakajima's problem: convex bodies of constant width and constant brightness

For a convex body $K\subset\R^n$, the $k$th projection function of $K$ assigns to any $k$-dimensional linear subspace of $\R^n$ the $k$-volume of the orthogonal projection of $K$ to that subspace. Let $K$ and $K_0$ be convex bodies in $\R^n$, and let $K_0$ be centrally symmetric and satisfy a weak regularity and curvature condition (which includes all $K_0$ with $\f K_0$ of class $C^2$ with positive radii of curvature). Assume that $K$ and $K_0$ have proportional 1st projection functions (i.e., width functions) and proportional $k$th projection functions. For $2\le k<(n+1)/2$ and for $k=3, n=5$ we show that $K$ and $K_0$ are homothetic. In the special case where $K_0$ is a Euclidean ball, we thus obtain characterizations of Euclidean balls as convex bodies of constant width and constant $k$-brightness.