标题： 穿透凸集 显示英文标题

标题： Piercing convex sets

摘要： 如果在该族的任何$p$个成员中，有些$q$有非空交集，则该集合族具有$(p,q)$属性。 证明了对于每个$p\ge q\ge d+1$，存在一个$c=c(p,q,d)<\infty$，使得对于每个在$R^d$中具有$(p,q)$性质的紧凸集族$\scr F$，存在一个至多包含$c$个点的集合，在$R^d$中，该集合与$\scr F$的每个成员相交。 这扩展了赫利定理并解决了哈迪格和德布纳的一个老问题。 显示英文摘要

摘要： A family of sets has the $(p,q)$ property if among any $p$ members of the family some $q$ have a nonempty intersection. It is shown that for every $p\ge q\ge d+1$ there is a $c=c(p,q,d)<\infty$ such that for every family $\scr F$ of compact, convex sets in $R^d$ that has the $(p,q)$ property there is a set of at most $c$ points in $R^d$ that intersects each member of $\scr F$. This extends Helly's Theorem and settles an old problem of Hadwiger and Debrunner.