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On generalized “ham sandwich” theorems. (English) Zbl 1164.58312

The author generalizes the “ham sandwich Theorem” to:
Let \(A_1,\ldots ,A_m\subseteq \mathbb R^n\) be subsets with finite Lebesgue measure. Then, for any sequence \(f_{0},\ldots ,f_{m}\) of \(\mathbb R\)-linearly independent polynomials in the polynomial ring \(\mathbb R[X_1,\ldots ,X_n]\) there are real numbers \(\lambda _0,\ldots ,\lambda _m\), not all zero, such that the real affine variety \(\{x\in \mathbb R^n;\;\lambda _{0}f_{0}(x)+\cdots +\lambda _{m}f_{m}(x)=0\}\) simultaneously bisects each of subsets \(A_k\), \(k=1,{\ldots },m\).
Some applications of the theorem are given — e.g.an answer to the question “which curves or manifolds other than straight lines or hyperplanes can serve as common medians for random vectors”, posed in [T.P.Hill, Am.Math.Mon.95, 437–441 (1988; Zbl 0643.60011)].
The proof of the generalized theorem is based on the famous Borsuk-Ulam Antipodal Theorem (K.Borsuk [Fundam.Math.20, 177–190 (1933; Zbl 0006.42403)]).

MSC:

58C07 Continuity properties of mappings on manifolds
12D10 Polynomials in real and complex fields: location of zeros (algebraic theorems)
14P05 Real algebraic sets
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