Nakajima’s problem for general convex bodies

Abstract:

For a convex body $K\subset \mathbb {R}^n$, the $k$th projection function of $K$ assigns to any $k$-dimensional linear subspace of $\mathbb {R}^n$ the $k$-volume of the orthogonal projection of $K$ to that subspace. Let $K$ and $K_0$ be convex bodies in $\mathbb {R}^n$, and let $K_0$ be centrally symmetric and satisfy a weak regularity assumption. Let $i,j\in \mathbb {N}$ be such that $1\le i<j\le n-2$ with $(i,j)\neq (1,n-2)$. Assume that $K$ and $K_0$ have proportional $i$th projection functions and proportional $j$th projection functions. Then we show that $K$ and $K_0$ are homothetic. In the particular case where $K_0$ is a Euclidean ball, we thus obtain characterizations of Euclidean balls as convex bodies having constant $i$-brightness and constant $j$-brightness. This special case solves Nakajima’s problem in arbitrary dimensions and for general convex bodies for most indices $(i,j)$.