A Problem of Geometry in R n

Abstract

AssTRAcr. Let 6f be a finite family of at least n + 1 convex sets in the n-dimensional Eucidean space Rn. Helly's theorem asserts that if all the (n + l)-subfamilies of IF have nonempty intersection, then f also has nonempty intersection. The main result in this paper is that if almost all of the (n + l)-subfamilies of Ti have nonempty intersection, then fS has a subfamily with nonempty intersection containing almost all of the sets in T.