  For a convex body $K\subset\R^n$ and $i\in\{1,...,n-1\}$, the function
assigning to any $i$-dimensional subspace $L$ of $\R^n$, the $i$-dimensional
volume of the orthogonal projection of $K$ to $L$, is called the $i$-th
projection function of $K$. Let $K, K_0\subset \R^n$ be smooth convex bodies of
class $C^2_+$, and let $K_0$ be centrally symmetric. Excluding two exceptional
cases, that of $(i,j)=(1,n-1)$ and $(i,j)=(n-2,n-1)$, we prove that $K$ and
$K_0$ are homothetic if they have two proportional projection functions. The
special case when $K_0$ is a Euclidean ball provides an extension of Nakajima's
classical three-dimensional characterization of spheres to higher dimensions.
