16 Touching simplices
Two \(d\)-simplices touch if their interiors are disjoint and their intersection is nonempty and has affine dimension \(d-1\). Let \(f(d)\) be the largest size of a pairwise touching family in \(\mathbb {R}^d\).
Bagemihl conjectured that for every \(d\ge 1\), we have \(f(d)=2^d\).
For every \(d\ge 2\), there is a family of \(2^d\) pairwise touching \(d\)-simplices in \(\mathbb {R}^d\) together with a transversal line that hits the interior of every single one of them.
Start with four touching triangles in the plane with a common transversal. For the induction step, an affine transformation puts short segments inside all simplices into a half-square. Reflect the configuration in \(x_1=x_2\), then form pyramids over the two configurations with apexes on opposite sides of \(\mathbb {R}^d\). This doubles the number of simplices. A slightly tilted antidiagonal meets the interior of each pyramid.
For all \(d \ge 1\), we have \(f(d){\lt} 2^{d+1}\).
Let \(s\) be the number of facet hyperplanes and put \(k=d+1\). Completing the zero entries of each row of the sign matrix gives \(2^{s-k}\) sign vectors. These sets of completions are pairwise disjoint, and at least one of the \(2^s\) possible sign vectors is missing. Thus \(r2^{s-k}{\lt}2^s\), so \(r{\lt}2^k=2^{d+1}\).