Helly-type theorems on common supporting lines for nonoverlapping families of convex bodies in the plane
Let us say that a family \({\mathbf F}\) of convex bodies has the property \(S\) if some line supports all the bodies, and the property \(D\) if in addition all the bodies are on the same side of the line. If every \(n\)-element subfamily of \({\mathbf F}\) has the property \(S\) (resp. \(D)\) we say that \({\mathbf F}\) has the property \(S(n)\) (resp. \(D(n))\). A Helly-type theorem on common supporting lines typically asserts that, for some class of family, \(S(n) \Rightarrow S\) or \(D(n)\Rightarrow D\). Typically, the configurations of bodies which prevent the Helly number from being lowered have rather few members; and restricting \({\mathbf F}\) to contain more than a certain number of bodies may allow \(n\) to be reduced. The first theorems of this type were obtained for families of disjoint bodies. In this paper, the authors consider families of bodies that are not necessarily disjoint, but are nonoverlapping -- that is, with pairwise disjoint interiors. They show that for nonoverlapping families of convex bodies, as for disjoint families, \(D(4)\Rightarrow D\), and give an example to show that (in contrast with the disjoint case) there is no threshold above which \(D(3)\Rightarrow D\). It is shown that a nonoverlapping family may have the property \(S(n)\) but not \(S\) for any \(n\). However, unless there is a point common to all the bodies, \(S(6)\Rightarrow S\). Moreover, if no \(k\) of the bodies have a common point, and there are at least \(12k-13\) bodies, \(S(5)\Rightarrow S\); and if there are at least \(72k-75\) bodies, \(S(4)\Rightarrow S\).
scientific article; zbMATH DE number 1374910