A New Topological Helly Theorem and some Transversals Results

We prove that for a topological space X with the property that $H_p(U)=0$ for $p\geq d$ and every open subset $U$ of $X$, a finite family of open sets in $X$ has nonempty intersection if for any subfamily of size $j$, $1\leq j \leq d+1$, 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.

No reviews yet.

No comments yet.