Let $ℱ=F_{\alpha }$ be a uniformly bounded collection of compact convex sets in ℝ ⁿ. Katchalski extended Helly’s theorem by proving for finite ℱ that dim (⋂ ℱ) ≥ d, 0 ≤ d ≤ n, if and only if the intersection of any f(n,d) elements has dimension at least d where f(n,0) = n+1 = f(n,n) and f(n,d) = maxn+1,2n-2d+2 for 1 ≤ d ≤ n-1. An equivalent statement of Katchalski’s result for finite ℱ is that there exists δ > 0 such that the intersection of any f(n,d) elements of ℱ contains a d-dimensional ball of measure...