The main result is a direct proof of the implication $(LVKF_{ k,3})\Rightarrow( LT_{ 3k-1,3})$ below. Consider the following statements: ($LVKF_{ 1,3}$) From any 11 points in $ \mathbb{ R}^{ 3}$ one can choose 3 pairwise disjoint triples whose convex hulls have a common point. ($LVKF_{ k,3}$) From any $6k + 5$ points in $ \mathbb{ R}^{ 3k}$ one can choose 3 pairwise disjoint sets each containing $2k + 1 $ points and whose convex hulls have a common point. ($LT_{ 2,3}$) Any 7 points in $\mathbb{ R}^{ 2}$ can be decomposed into 3 subsets whose convex hulls have a common point. ($LT_{ d,3}$) Any $2d+3$ points in $\mathbb{ R}^d$ can be decomposed into 3 subsets whose convex hulls have a common point. This statements are true, but the meaning of the article is the direct derivation of one statement from another.
Categories: math.GT