The Katchalski-Lewis transversal problem in \(\mathbb R^{n}\) (Q878386)

Let \(F\) be a family of disjoint translates of a convex set in \({\mathbb R}^n\). The family \(F\) is said to have the \(T_k(m)\) (resp. \(T_k\)) property if every \(m\) members (resp. all members) of \(F\) are intersected by a \(k\)-dimensional plane. In [Proc. Am. Math. Soc. 79, 457--461 (1980; Zbl 0435.52006)], \textit{M. Katchalski} and \textit{T. Lewis} showed that, for a family \(F\) of disjoint translates of a planar convex set in \({\mathbb R}^2\), there is a constant \(c\) independent of \(F\) such that if \(F\) has the \(T_1(3)\) property then there is a subfamily \(G\subset F\), excluding at most \(c\) members of \(F\), which has the \(T_1\) property. In the present paper the author proves an extension of this result to arbitrary dimension: for \(n\geq 2\) there exists a constant \(c(n)\) such that if \(F\) has the \(T_1(3)\) property then there is a subfamily \(G\subset F\), excluding at most \(c(n)\) members of \(F\), which has the \(T_{n-1}\) property. The theorem is obtained as a consequence of a result of the same type in the plane, since it is also proved that the \(T_1(3)\) property is preserved under projections. The last two sections are devoted to study the case of the Euclidean ball. Two particular families of translates of the unit ball in \({\mathbb R}^3\) are described, which provide counterexamples to some related questions.