For any collection of three solids in the three-dimensional space there exists a plane which simultaneously bisects all of them, i.e. divides each of the solids into two halfs of equal volume. In a popular form this result is stated as the fact that it is possible to cut fairly an open ham-sandwich consisting of two pieces of bread and a piece of ham with a single straight cut. More generally, for a collection of measurable sets (mass distributions, finite sets) in there exists a hyperplane simultaneously bisecting all of them.
The ham-sandwich theorem is a consequence of the well-known Borsuk–Ulam theorem, which says that for any continuous mapping from a -dimensional sphere into , there exists a pair of antipodal points such that . (Cf. also Antipodes.)
Other examples of combinatorial partitions of masses include the "centre-point theorem" and the related "centre-transversal theorem" , equi-partitions of masses by higher-dimensional "orthants" , equi-partitions by convex cones, partitions of lines and other geometric objects, etc.
The centre-point theorem says that for any measurable set in there exists a point such that for any half-space : if then
The centre-transversal theorem, [a5], is a generalization of both the ham-sandwich and the centre-point theorem and it claims that for any collection , , of Lebesgue-measurable sets in (cf. also Lebesgue measure) there exists a -dimensional affine subspace such that for every closed half-space and every : if then
An example of an equi-partition result into higher-dimensional "orthants" is as follows, [a3]. Any measurable set can be partitioned into pieces of equal measure by hyperplanes.
The ham-sandwich theorem, together with other relatives belonging to combinatorial (equi)partitions of masses, has been often applied to problems of discrete and computational geometry, see [a5] for a survey.
See also Comitant.
|[a1]||A. Björner, "Topological methods" R. Graham (ed.) M. Grötschel (ed.) L. Lovász (ed.) , Handbook of Combinatorics , North-Holland (1995)|
|[a2]||E. Fadell, S. Husseini, "An ideal-valued cohomological index theory with applications to Borsuk–Ulam and Bourgin–Yang theorems" Ergod. Th. & Dynam. Sys. , 8 (1988) pp. 73–85|
|[a3]||E. Ramos, "Equipartitions of mass distributions, by hyperplanes" Discr. Comp. Geometry , 15 (1996) pp. 147-167|
|[a4]||H. Steinlein, "Borsuk's antipodal theorem and its generalizations, and applications: a survey" , Topological Methods in Nonlinear Analysis , Sém. Math. Sup. , 95 , Presses Univ. Montréal (1985) pp. 166–235|
|[a5]||R.T. Živaljević, "Topological methods" J.E. Goodman (ed.) J. O'Rourke (ed.) , CRC Handbook of Discrete and Combinatorial Geometry , CRC Press (1997)|
|[a6]||R.T. Živaljević, "User's guide to equivariant methods in combinatorics" Publ. Inst. Math. Belgrade , 59 (73) (1996)|
Ham-sandwich theorem. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Ham-sandwich_theorem&oldid=42200