Jesús Jerónimo-Castro

About the piercing number of a family of intervals

In this paper, we develop a Helly-Gallai type theorem for piercing number on finite families of closed intervals in R d , as well as some bounds for the piercing number of families of lines and intervals satisfying the ( p , 3 ) -property.

T(4) Families of {\varphi }-Disjoint Ovals

Let K denote an oval, a centrally symmetric compact convex domain with non-empty interior. A family of translates of K is said to have property T(k) if, for every subset of at most k translates, there exists a common line transversal intersecting all of them. Property T means that there is a line transversal to all the members of the family. Two translates, \(K_i\) and \(K_j\) of K, are said to be \(\varphi \)-disjoint, \(\varphi > 0\), if the concentric \(\varphi \)-enlarged copies of \(K_i\) and \( K_j\) are disjoint. It is known [5] that in a finite \(\varphi \)-disjoint family of translates of an oval, \(T(3) \Rightarrow T\) if \(\varphi >{2}\), and this bound is sharp. In this note, finite \(\varphi \)-disjoint T(4)-families of translates of an oval will be investigated.