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<p>A theorem of Grünbaum, which states that every <i>m</i>-polytope is an affine projection of an <i>m</i>-simplex (namely, combinatorially, the faces of the polytope form a refinement of the faces of the simplex), implies the following generalization of Tverberg’s theorem: if <i>f</i> is a linear function from an <i>m</i>-dimensional polytope <i>P</i> to <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {R}^d\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(m \ge (d + 1)(r - 1)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>≥</mo> <mo stretchy="false">(</mo> <mi>d</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, then there are <i>r</i> pairwise disjoint faces of <i>P</i> whose images intersect. Moreover, the topological Tverberg theorem implies that this statement is true whenever the map <i>f</i> is continuous and <i>r</i> is a prime power. In this note we show that for certain families of polytopes the lower bound on the dimension <i>m</i> of the polytopes can be significantly improved, both in the affine and topological cases.</p>

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Improved Tverberg Theorems for Certain Families of Polytopes

  • 期刊论文

  • 发表日期：

    2026年7月9日

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  • Pablo Soberón,
  • Shira Zerbib

摘要

A theorem of Grünbaum, which states that every m-polytope is an affine projection of an m-simplex (namely, combinatorially, the faces of the polytope form a refinement of the faces of the simplex), implies the following generalization of Tverberg’s theorem: if f is a linear function from an m-dimensional polytope P to \(\mathbb {R}^d\) R d and \(m \ge (d + 1)(r - 1)\) m ≥ ( d + 1 ) ( r - 1 ) , then there are r pairwise disjoint faces of P whose images intersect. Moreover, the topological Tverberg theorem implies that this statement is true whenever the map f is continuous and r is a prime power. In this note we show that for certain families of polytopes the lower bound on the dimension m of the polytopes can be significantly improved, both in the affine and topological cases.