Ham sandwich theorem

In n-dimensional space, a hyperplane can simultaneously bisect n finite measures. In three dimensions, one plane can cut three objects so each piece has half the measure of each object.

Borsuk–Ulam theorem: Every continuous map from an n-sphere to n-dimensional Euclidean space identifies some pair of antipodal points. Its antipodal-point guarantee yields a hyperplane that bisects all n measures.
Lebesgue measure: A standard measure assigning lengths, areas, and volumes to suitable subsets of Euclidean space. It models ordinary geometric volume when objects are treated as regions.
Computational geometry: The study of algorithms and structures for geometric problems. Algorithms seek ham-sandwich cuts for point sets and other represented data.
Two-dimensional ham-sandwich theorem: The result that a line can bisect two finite measures in the plane. It is the planar instance, where one line balances two distributions.
Hugo Steinhaus: A Polish mathematician known for work in analysis, probability, and mathematical culture. He is associated with the early formulation that inspired the theorem's sandwich name.
Hyperplane: A flat subspace with dimension one less than its ambient space. The theorem's bisecting cut is a hyperplane, whose orientation and position vary continuously.
Absolutely continuous measure: A measure that assigns zero to every set having zero measure under a specified reference measure. The usual theorem assumes measures without mass concentrated on lower-dimensional sets.
Ham-sandwich cut: A hyperplane that bisects each of several measures or finite point sets. It is the geometric object computed or constructed using the theorem.
Three-dimensional ham-sandwich theorem: The result that a plane can bisect three finite measures in three-dimensional space. It gives the familiar example of one cut halving three objects.
Stefan Banach: A Polish mathematician who helped establish modern functional analysis. He appears in the early history of the named bisection problem.

Linked from 4 pages