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 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 Sθ denote the (n − 1)-dimensional hyperplane containing θ and S. Define r(θ) to be the (n − 1)-dimensional volume of K ∩ Sθ. Let C be the curve {θr(θ)} in S⊥. Then C forms the boundary of a convex body in S⊥.

See also Brunn–Minkowski inequality Prékopa–Leindler inequality References Busemann, Herbert (1949). "A theorem on convex bodies of the Brunn-Minkowski type". Proc. Natl. Acad. Sci. U.S.A. 35 (1): 27–31. doi:10.1073/pnas.35.1.27. PMC 1062952. PMID 16588849. Gardner, Richard J. (2002). "The Brunn-Minkowski inequality". Bull. Amer. Math. Soc. (N.S.). 39 (3): 355–405 (electronic). CiteSeerX 10.1.1.106.7344. doi:10.1090/S0273-0979-02-00941-2. Categories: Euclidean geometryGeometric inequalitiesTheorems in convex geometry