Busemann's theorem

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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.

[edit] 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. Given a unit vector θ in S^⊥, the orthogonal complement of S, let S_θ denote the closed (n − 1)-dimensional half-space containing θ and with S as its boundary. Let r = r(θ) be the curve in S^⊥ such that r(θ) ≥ 0 is the (n − 1)-dimensional volume of K ∩ S_θ. Then r forms the boundary of a convex body in S^⊥.

[edit] See also

• Busemann-Barthel-Franz inequality
• Prékopa-Leindler inequality

[edit] References

• Busemann, Herbert (1949). "The isoperimetric problem for Minkowski area". Amer. J. Math. 71: 743–762. doi:10.2307/2372362. 
• 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.