Helly-type theorems on the homology of the space of transversals
An \(n\)-dimensional flat \(P\) is a \(n\)-transversal of a family \({\mathcal C}= \{K_i\}_{i=0}^d\) of convex bodies in \({R}^{n+k}\) if \(P\) intersects \(K_i\) for each \(i=0,\ldots, d\). Let \(AG_n(R^{n+k})\), respectively \(G_n(R^{n+k})\), be the Grassmannian of all affine, respectively linear, subspaces in \(R^{n+k}\). Let \(T_n({\mathcal C})\subset AG_n(R^{n+k})\) be the space of all \(n\)-transversals of \({\mathcal C}\). The authors introduce a homological measure of complexity \(\mu(X)\) for subsets \(X\subset AG_n(R^{n+k})\subset G_{n+1}(R^{n+k+1})\) and use it to formulate Helly-type results in the spirit of classical results of Horn and Klee. Informally speaking \(\mu(X)\geq r\) if \(X\) contains a ``translate'' of the fundamental cycle of \(G_n(R^{n+r})\). The main result says that if \(1\leq m\leq n\) then \(\mu(T_n({\mathcal C}))\geq k\) if and only if \(\mu(T_m({\mathcal C}')\geq k\) for each \({\mathcal C}'\subset {\mathcal C}\) of size \(k+2\). For example if \(m,k=1\) and \(n=2\), the result says that each three convex sets in \({\mathcal C}\) admit a non-contractible loop of transversal lines if and only if \({\mathcal C}\) has transversal planes orthogonal to every direction prescribed in advance.
scientific article; zbMATH DE number 1747948