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"⊥.