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.

tatement 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". 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"^⊥.

ee also

* Busemann-Barthel-Franz inequality
* Prékopa-Leindler inequality

References

* cite journal
last = Busemann
first = Herbert
title = The isoperimetric problem for Minkowski area
journal = Amer. J. Math.
volume = 71
year = 1949
pages = 743–762
doi = 10.2307/2372362
* cite journal
last=Gardner
first=Richard J.
title=The Brunn-Minkowski inequality
journal=Bull. Amer. Math. Soc. (N.S.)
volume=39
issue=3
year=2002
pages=355–405 (electronic)
doi=10.1090/S0273-0979-02-00941-2