Smooth convex Bodies with proportional projection functions

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For a convex body K⊂ⁿ and i∈{1,...,n-1}, the function assigning to any i-dimensional subspace L of ⁿ, the i-dimensional volume of the orthogonal projection of K to L, is called the i-th projection function of K. Let K, K₀⊂ ⁿ be smooth convex bodies of class C²+, and let K₀ 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 K₀ are homothetic if they have two proportional projection functions. The special case when K₀ is a Euclidean ball provides an extension of Nakajima's classical three-dimensional characterization of spheres to higher dimensions.