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Common hyperplane medians for random vectors. (English) Zbl 0643.60011

Every collection of \(m\leq n\) random \(n\)-dimensional vectors has a common hyperplane median, that is, there is a single hyperplane which simultaneously bisects each of the distributions in the sense that at least half the mass of each distribution lies on one side of the hyperplane (including the hyperplane), and at least half lies on the other side (again including the hyperplane).
This result generalizes the well-known Ham Sandwich Theorem; the proof is based on first principles using an application of the Borsuk-Ulam Theorem [e.g.: K. Borsuk, Fundam. Math. 20, 177-190 (1933; Zbl 0006.42403)] to a “midpoint median” function.
Reviewer: P.T.Hill

MSC:

60D05 Geometric probability and stochastic geometry
28B05 Vector-valued set functions, measures and integrals

Citations:

Zbl 0006.42403
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