A New Topological Helly Theorem and Some Transversal Results

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• 作者：Luis Montejano (1)
  
• 关键词：Helly theorem ; Homology group ; Transversal ; Primary 52A35 ; 55N10
• 刊名：Discrete and Computational Geometry
• 出版年：2014
• 出版时间：September 2014
• 年：2014
• 卷：52
• 期：2
• 页码：390-398
• 全文大小：138 KB
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  8. Debrunner, H.E.: Helly type theorems derived from basic singular homology. Am. Math. Mon. 17(4), 375鈥?80 (1970) 10.2307/2316144" target="_blank" title="It opens in new window">CrossRef
  9. Eckhoff, J.: Helly, Radon and Carath茅odory type theorems. In: Gruber, P.M., Wills, J.M. (eds.) Handbook of Convex Geometry. North-Holland, Amsterdam (1993)
  10. Kalai, G., Meshulam, R.: Leray numbers of projections and a topological Helly type theorem. J. Topol. 1, 551鈥?56 (2008) 10.1112/jtopol/jtn010" target="_blank" title="It opens in new window">CrossRef
  11. Pontryagin, L.S.: Characteristic cycles on differential manifolds. Transl. Am. Math. Soc. 32, 149鈥?18 (1950)
  
• 作者单位：Luis Montejano (1)
  
  1. Instituto de Matem谩ticas, Unidad Juriquilla, Universidad Nacional Aut贸noma de M茅xico, Mexico City, Mexico
  
• ISSN：1432-0444

文摘

We prove that for a topological space \(X\) with the property that \( H_{*}(U)=0\) for \(*\ge d\) and every open subset \(U\) of \(X\) , a finite family of open sets in \(X\) has nonempty intersection if for any subfamily of size \(j,\,1\le j\le d+1,\) the \((d-j)\) -dimensional homology group of its intersection is zero. We use this theorem to prove new results concerning transversal affine planes to families of convex sets.