VC dimension of intersection of half-spaces Define $$l_i(x) := \text{sgn} \left( w_i^\top x - b_i \right)$$

for $i=1,...,n$, where $x \in \mathbb{R}^d$.

Then define the classifier

$$ g(x) := \max \{ l_1(x),..., l_n(x) \}$$

which represents the intersection of $n$ linear functions and hence a convex polytope in $\mathbb{R}^d$ with $n$ facets.

What is the VC dimension of the family of all $g$ of this form?

I found that an upper bound is $2 (d+1) n \text{log}_2( (d+1)n ).$
It has been recently shown by Csikos, Kupavskii, Mustafa
in "Optimal Bounds on the VC-dimension" that the VC dimension of $k$-fold unions (or intersections or XORs) of half-spaces in $R^d$ behaves as
$$ \Theta(dk\log k).$$