Borsuk–Ulam theorem

Every continuous map from the n-dimensional sphere to n-dimensional Euclidean space takes equal values at some pair of antipodal points. It holds in every positive dimension and has equivalent formulations involving odd maps.

Antipodal map: A map on a sphere that sends each point to its opposite point. The theorem’s central condition pairs points exchanged by this map.
Ham sandwich theorem: A theorem guaranteeing one hyperplane that simultaneously bisects each of several measurable objects. Its standard proof uses Borsuk–Ulam to find a bisecting hyperplane.
Karol Borsuk: A Polish mathematician known for foundational work in topology and shape theory. His 1933 paper established the theorem’s modern form.
Brouwer fixed-point theorem: Every continuous map from a closed ball to itself has a fixed point. It concerns a fixed point in a ball, whereas Borsuk–Ulam concerns equal values at opposite sphere points.
Circle: The set of points in a plane at a fixed distance from a center. For a circle mapped continuously to the real line, some opposite pair has equal values.
Odd function: A function f satisfying f(−x) = −f(x) wherever both inputs are defined. Subtracting a map’s values at antipodes produces an odd map in equivalent formulations.
Neutrino mass measurement: Experimental methods for estimating neutrino masses from particle observations. A classic consequence says two antipodal points on Earth share temperature and pressure, illustrating paired measurements.
Stanislaw Ulam: A Polish-American mathematician who made major contributions to set theory, probability, and mathematical physics. The theorem bears his name alongside Borsuk’s.
Invariance of domain: A theorem stating that continuous injective maps between equal-dimensional Euclidean spaces have open images. Both constrain continuous maps by dimension, but only Borsuk–Ulam forces a collision.
Sphere: The set of points at a fixed distance from a center in Euclidean space. The theorem applies to spheres in every positive dimension.

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