Bounded boxes, Hausdorff distance, and a new proof of an interesting Helly-type theorem

We give a short proof of the theorem that any family of subsets ofRd, with the property that the intersection of any nonempty finite subfamily can be represented as the disjoint union of at mostk closed convex sets, has Helly number at mostk(d+1).

We introduce a new variant of quantitative Helly-type theorems: the minimal homothetic distance of the intersection of a family of convex sets to the intersection of a subfamily of a fixed size. As an application, we establish the following quantitative ...

We prove that for a topological space $$X$$ X with the property that $$ H_{*}(U)=0$$ H ( U ) = 0 for $$*\ge d$$ d and every open subset $$U$$ U of $$X$$ X , a finite family of open sets in $$X$$ X has nonempty intersection if for any subfamily of size $$j,\,1\le j\le d+1,$$ j , 1 ≤ j ≤ d + 1 , the $$(d-j)$$ ( d - j ) -dimensional homology group of its intersection is zero. We use ...