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Borsuk-Ulam Theorem

The Borsuk-Ulam theorem states that every continuous map f:S^n->R^n must identify a pair of antipodal points.

Ullrich (2026) proved an adaptive-measurement extension for the sphere S_R={x in R^m:||x||=R}, R>0, of any norm on R^m. An adaptive measurement may choose its next continuous real-valued function from the previous measurement values. If 2^n<=m, then any n such measurements take identical values on some antipodal points x,-x of S_R. Consequently, approximating every point of the radius-R ball with worst-case error strictly less than R requires at least [log_2(m+1)] measurements, where [x] is the ceiling function. The sphere S_R has dimension m-1. Ullrich (2026) reports AI assistance with the mathematical proof.

See also

Antipodal Points

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References

Dodson, C. T. J. and Parker, P. E. A User's Guide to Algebraic Topology. Dordrecht, Netherlands: Kluwer, pp. 121 and 284, 1997.Ullrich, M. "A Borsuk-Ulam Theorem for Adaptive Continuous Measurements." 24 Oct 2026. https://arxiv.org/abs/2610.33781.

Referenced on Wolfram|Alpha

Borsuk-Ulam Theorem

Cite this as:

Weisstein, Eric W. "Borsuk-Ulam Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Borsuk-UlamTheorem.html

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