A New Topological Helly Theorem and Some Transversal Results
نویسنده
چکیده
We prove that for a topological spaceX with the property that H∗(U) = 0 for ∗ ≥ 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 ≤ j ≤ d+1, the (d−j)-dimensional homology group of its intersection is zero. We use this theorem to prove new results concerning transversal affi ne planes to families of convex sets
منابع مشابه
Topological transversals to a family of convex sets
Let F be a family of compact convex sets in R. We say that F has a topological ρ-transversal of index (m, k) (ρ < m, 0 < k ≤ d − m) if there are, homologically, as many transversal m-planes to F as m-planes containing a fixed ρ-plane in R. Clearly, if F has a ρ-transversal plane, then F has a topological ρ-transversal of index (m, k), for ρ < m and k ≤ d − m. The converse is not true in general...
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ورودعنوان ژورنال:
- Discrete & Computational Geometry
دوره 52 شماره
صفحات -
تاریخ انتشار 2014