Smooth convex Bodies with proportional projection functions

Abstract

For a convex body $K\subset {\mathbb{R}}^n$ and $i\in \{1,...,n-1\}$, the function assigning to any $i$-dimensional subspace $L$ of ${\mathbb{R}}^n$, the $i$-dimensional volume of the orthogonal projection of $K$ to $L$, is called the $i$-th projection function of $K$. Let $K,K_0\subset {\mathbb{R}}^n$ 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.

17 pages