Consider a fixed set of vectors $\{x_i\}_{i\in[n]}$ in $\mathbb R^d$ and closed polyhedral cone $C = \{w \in \mathbb R^d : w^\top x_i \geq 0, \forall i \in [n]\}$ with full dimension i.e. $C$ contains a set of $d$ independent vectors. For each $w\in C$, the corresponding labeling hypothesis is given by $h_w(x) = {1}[w^\top x\geq 0]$ where $1[\cdot]$ is an indicator function which outputs 1 if its argument evaluates to true and 0 otherwise. What is the VC dimension of $C$ and how would one go about proving it?
P.S. I am aware of this answer where the author @domotorp suggest that VC should be $d-1$ for any full dimensional cone but their suggestion of replacing set of $e_i$'s by any set of vector(or even any set of $d$ independent vectors) whose negation is in the cone does not seem to work. More concretely, it is not clear if we choose to shatter a set of independent points $\{v_i\}|_{i\in[d]}$ such that $\forall i \in [d], -v_i \in C$, what are the hypotheses in $C$ that should produce a label vector $l$ for all $l \in \{0,1\}^d\backslash\{0^d, 1^d\}$? It seems like some sort of Seperating hyperplane theorem and Radon theorem for cone may help but how exactly is not very clear to me.
