Note on a conjecture of Sierksma (Q1802219)

In extension of Tverberg's theorem, the maximum \(S(q,d)\) of all numbers \(\nu\) is considered such that, for every linear map \(h\) from the simplex \(\Delta^{(q-1)(d+1)}\) into \(\mathbb{R}^ d\) (which has the property that the images of the vertices are in general position), there exist at least \(\nu\) different collections \(\{\Delta^{t_ 1}, \ldots, \Delta^{t_ q}\}\) of disjoint faces of \(\Delta^{(q-1)(d+1)}\) with \(h(\Delta^{i_ 1})\cap \cdots\cap h( \Delta^{t_ q}) \neq \emptyset\). Sierksma's conjecture is that \(S(q,d)=\bigl( (q-1)! \bigr)^ d\). The main result is that \[ S(q,d) \geq {1\over (q-1)!} \left( {q \over 2} \right)^{((q- 1)(d+1))/2}, \] in case \(q \geq 3\) is a prime number. The result also holds for (general position) continuous maps \(f:\Delta^{(q-1)(d+1)} \to \mathbb{R}^ d\).