Busemann's theorem
# 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.

## Statement of the theorem

Let K be a convex body in n-dimensional Euclidean space $R^n$ containing the origin in its interior. Let S be an (n − 2)-dimensional linear subspace of $R^n.$ 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 $S^⊥.$

