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 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^⊥.

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. PMID 16588849. 
• Gardner, Richard J. (2002). "The Brunn-Minkowski inequality". Bull. Amer. Math. Soc. (N.S.) 39 (3): 355–405 (electronic). doi:10.1090/S0273-0979-02-00941-2.