Pizza theorem

The pizza theorem states that four lines through a disk’s center divide it into eight regions whose alternating areas are equal, under specified conditions on the cuts.

Area: A measure of the size of a two-dimensional region. The theorem compares sums of areas rather than the shapes of the slices.
Upton's pizza theorem: An area-equality theorem for alternating pieces formed by radial cuts through a disk. This alternate name links the result to its popular mathematical presentation.
Line: A straight path extending infinitely in both directions in Euclidean geometry. Each cut is part of a line passing through the disk’s center.
Fair division: The study of procedures for dividing resources among participants according to fairness criteria. The pizza result offers a geometric example of equal-valued shares.
Generalized pizza theorem: Extensions of the pizza theorem to broader classes of centrally symmetric regions or cutting arrangements. Generalizations test how far the alternating-area equality survives beyond circular disks.
Disk: The planar region consisting of a circle and all points inside it. The theorem’s equal-area result applies to a disk cut through its center.
Michael Spivak: An American mathematician known for textbooks on calculus, differential geometry, and analysis. His exposition helped establish the theorem as a memorable calculus example.
Euclidean geometry: The geometry of points, lines, distances, and angles on a flat plane. The cuts and disk are described using ordinary plane geometry.
Cake-cutting problem: A family of problems about dividing a divisible resource fairly among participants. Its fairness questions extend beyond the theorem’s particular alternating slices.
Convex body: A compact convex set in a real vector space. The disk is one centrally symmetric convex body among possible settings for extensions.

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Borsuk–Ulam theoremCompared with: It is another geometric coincidence result, but its proof relies on area symmetry rather than antipodal topology.