A Helly-type theorem for higher-dimensional transversals (Q5959560)

A \(k\)-dimensional flat \(P\) is a \(k\)-transversal of a family \(\mathcal{C}\) of convex bodies in \({R}^d\) if \(P\) intersects each \(K\in \mathcal{C}\). L.~Santaló showed that there does not exist a Helly-type theorem for transversals of positive dimension to unrestricted collections of convex bodies. This focused the research on families \(\mathcal{C}\) satisfying some restriction on the size or the shape of its elements. Danzer, Grünbaum and Klee in 1963, after a preliminary result of Hadwiger, proved that a Helly-type result exists for \(1\)-transversals, with Helly number \(d+1\), if the convex bodies are of bounded diameter while the diameter of the union \(\bigcup\mathcal{C}\) is infinite. NEWLINENEWLINENEWLINEThe authors extend the result of Danzer, Grünbaum and Klee to the case of \(k\)-transversals \((1\leq k\leq d-1)\). The condition that \(\bigcup\mathcal C\) is unbounded (the case \(k=1\)) is replaced by the condition that \(\bigcup\mathcal C\) is unbounded in `\(k\)-independent directions'. The authors show that the \(k\)-unboundedness condition is essential by constructing an example of a family unbounded in \((k-1)\) dimensions which does not admit a \(k\)-flat transversal.