A problem of geometry in $\textbf {R}^{n}$

Abstract:

Let $\mathcal {F}$ be a finite family of at least $n + 1$ convex sets in the n-dimensional Euclidean space ${R^n}$. Helly’s theorem asserts that if all the $(n + 1)$-subfamilies of $\mathcal {F}$ have nonempty intersection, then $\mathcal {F}$ also has nonempty intersection. The main result in this paper is that if almost all of the $(n + 1)$-subfamilies of $\mathcal {F}$ have nonempty intersection, then $\mathcal {F}$ has a subfamily with nonempty intersection containing almost all of the sets in $\mathcal {F}$.