View source for Ham-sandwich theorem

''Stone–Tukey theorem''

For any collection of three solids in the three-dimensional space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110060/h1100601.png" /> there exists a plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110060/h1100602.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110060/h1100603.png" /> measurable sets (mass distributions, finite sets) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110060/h1100604.png" /> there exists a hyperplane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110060/h1100605.png" /> 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|continuous mapping]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110060/h1100606.png" /> from a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110060/h1100607.png" />-dimensional sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110060/h1100608.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110060/h1100609.png" />, there exists a pair of antipodal points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110060/h11006010.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110060/h11006011.png" />. (Cf. also [[Antipodes|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|measurable set]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110060/h11006012.png" /> there exists a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110060/h11006013.png" /> such that for any half-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110060/h11006014.png" />: if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110060/h11006015.png" /> then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110060/h11006016.png" /></td> </tr></table>

The centre-transversal theorem, [[#References|[a5]]], is a generalization of both the ham-sandwich and the centre-point theorem and it claims that for any collection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110060/h11006017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110060/h11006018.png" />, of Lebesgue-measurable sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110060/h11006019.png" /> (cf. also [[Lebesgue measure|Lebesgue measure]]) there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110060/h11006020.png" />-dimensional affine subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110060/h11006021.png" /> such that for every closed half-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110060/h11006022.png" /> and every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110060/h11006023.png" />: if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110060/h11006024.png" /> then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110060/h11006025.png" /></td> </tr></table>

An example of an equi-partition result into higher-dimensional  "orthants"  is as follows, [[#References|[a3]]]. Any measurable set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110060/h11006026.png" /> can be partitioned into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110060/h11006027.png" /> pieces of equal measure by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110060/h11006028.png" /> hyperplanes.

The proofs of all these and other related results are topological and use several forms of generalized Borsuk–Ulam-type theorems, see [[#References|[a2]]], [[#References|[a4]]], [[#References|[a6]]].

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 [[#References|[a5]]] for a survey.

See also [[Comitant|Comitant]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Björner,   "Topological methods"  R. Graham (ed.)  M. Grötschel (ed.)  L. Lovász (ed.) , ''Handbook of Combinatorics'' , North-Holland  (1995)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E. Fadell,   S. Husseini,   "An ideal-valued cohomological index theory with applications to Borsuk–Ulam and Bourgin–Yang theorems"  ''Ergod. Th. &amp; Dynam. Sys.'' , '''8'''  (1988)  pp. 73–85</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E. Ramos,   "Equipartitions of mass distributions, by hyperplanes"  ''Discr. Comp. Geometry'' , '''15'''  (1996)  pp. 147-167</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  R.T. Živaljević,   "Topological methods"  J.E. Goodman (ed.)  J. O'Rourke (ed.) , ''CRC Handbook of Discrete and Combinatorial Geometry'' , CRC Press  (1997)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  R.T. Živaljević,   "User's guide to equivariant methods in combinatorics"  ''Publ. Inst. Math. Belgrade'' , '''59 (73)'''  (1996)</TD></TR></table>