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<p>We give a new proof of Tverberg’s famous theorem: For every set <i>X</i> ⊂ ℝ<sup><i>d</i></sup> with ∣<i>X</i>∣ = (<i>r</i>−1)(<i>d</i>+1) + 1, there is a partition of <i>X</i> into <i>r</i> sets <i>X</i><sub>1</sub>,…, <i>X</i><sub><i>r</i></sub> such that <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\bigcap\nolimits_{p=1}^r \text{conv}X_{p}\ne\emptyset\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msubsup> <mo movablelimits="false">⋂</mo> <mrow> <mi>p</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>r</mi> </msubsup> <mtext>conv</mtext> <msub> <mi>X</mi> <mrow> <mi>p</mi> </mrow> </msub> <mo>≠</mo> <mi mathvariant="normal">∅</mi> </math></EquationSource> </InlineEquation>. The new proof uses linear algebra, specially structured matrices, the theory of linear equations, and Tverberg’s original “moving the points” method.</p>

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Tverberg’s theorem, a new proof

  • 期刊论文

  • 发表日期：

    2025年12月11日

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  • Imre Bárány

摘要

We give a new proof of Tverberg’s famous theorem: For every set X ⊂ ℝd with ∣X∣ = (r−1)(d+1) + 1, there is a partition of X into r sets X1,…, Xr such that \(\bigcap\nolimits_{p=1}^r \text{conv}X_{p}\ne\emptyset\) ⋂ p = 1 r conv X p ≠ ∅ . The new proof uses linear algebra, specially structured matrices, the theory of linear equations, and Tverberg’s original “moving the points” method.