Busemann's theorem

In mathematics, Busemann's theorem is a theorem in Euclidean geometry and geometric tomography. It was first proved by Herbert Busemann in 1949 and was motivated by his theory of area in Finsler spaces.

Let K be a convex body in n-dimensional Euclidean space Rⁿ containing the origin in its interior. Let S be an (n − 2)-dimensional linear subspace of Rⁿ. For each unit vector θ in S^⊥, the orthogonal complement of S, let S_θ denote the (n − 1)-dimensional hyperplane containing θ and S. Define r(θ) to be the (n − 1)-dimensional volume of K ∩ S_θ. Let C be the curve {θr(θ)} in S^⊥. Then C forms the boundary of a convex body in S^⊥.

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