부산대학교 도서관

The author gives a metric analogue of the well-known Helly's theorem. Let $w(K)$ be the width of a closed convex set $K$ in $d$-dimensional Euclidean space $E^d$. Theorem: Let $K_1,\cdots,K_n$ be a collection of closed convex sets in $E^d$ with the property that the width of the intersection of every $d+1$ of them is at least $\alpha\geq 0$; then $w(K_1\cap\cdots\cap K_n)=\beta\geq\alpha$; moreover, there exists some $d+1$ of the $K_i$, say $K_1,\cdots,K_{d+1}$, such that $w(K_1\cap\cdots\cap K_{d+1})=\beta$. \par A somewhat weaker version of this theorem for infinite collections of convex sets and a new proof of Helly's theorem are also presented. \par \{For the entire collection see MR0388240 (52 \#9077).\}