McMullen problem

Gale transform

Using the Gale transform, this problem can be reformulated as:

Determine the smallest number ${\displaystyle \mu (d)}$ such that for every set of ${\displaystyle \mu (d)}$ points ${\displaystyle X=\{x_{1},x_{2},\dots ,x_{\mu (d)}\}}$ in linearly general position on the sphere ${\displaystyle S^{d-1}}$ it is possible to choose a set ${\displaystyle Y=\{\varepsilon _{1}x_{1},\varepsilon _{2}x_{2},\dots ,\varepsilon _{\mu (d)}x_{\mu (d)}\}}$ where ${\displaystyle \varepsilon _{i}=\pm 1}$ for ${\displaystyle i=1,2,\dots ,\mu (d)}$, such that every open hemisphere of ${\displaystyle S^{d-1}}$ contains at least two members of ${\displaystyle Y}$.

The numbers ${\displaystyle \nu }$ of the original formulation of the McMullen problem and ${\displaystyle \mu }$ of the Gale transform formulation are connected by the relationships $${\displaystyle {\begin{aligned}\mu (k)&=\min\{w\mid w\leq \nu (w-k-1)\}\\\nu (d)&=\max\{w\mid w\geq \mu (w-d-1)\}\end{aligned}}}$$

Partition into nearly-disjoint hulls

Also, by simple geometric observation, it can be reformulated as:

Determine the smallest number ${\displaystyle \lambda (d)}$ such that for every set ${\displaystyle X}$ of ${\displaystyle \lambda (d)}$ points in ${\displaystyle \mathbb {R} ^{d}}$ there exists a partition of ${\displaystyle X}$ into two sets ${\displaystyle A}$ and ${\displaystyle B}$ with $${\displaystyle \operatorname {conv} (A\backslash \{x\})\cap \operatorname {conv} (B\backslash \{x\})\not =\varnothing ,\forall x\in X.\,}$$

The relation between ${\displaystyle \mu }$ and ${\displaystyle \lambda }$ is $${\displaystyle \mu (d+1)=\lambda (d),\qquad d\geq 1\,}$$

Projective duality

The equivalent projective dual statement to the McMullen problem is to determine the largest number ${\displaystyle \nu (d)}$ such that every set of ${\displaystyle \nu (d)}$ hyperplanes in general position in d-dimensional real projective space form an arrangement of hyperplanes in which one of the cells is bounded by all of the hyperplanes.