Let $\mathcal{C}$ be a pointed, salient cone in $\mathbb{R}^d$. We may also assume that $\mathcal{C}$ is full-dimensional. Consider the set of binary classifiers $$\mathcal{H} = \{\boldsymbol{x}\mapsto\boldsymbol{1}\hspace{-0.9ex}\mathrm{1}(\boldsymbol{h}^\mathsf{T}\boldsymbol{x}\ge 0)\mid \boldsymbol{h} \in \mathcal{C} \}.$$ Is there any bound known for the VC-dimension of $\mathcal{H}$ that depends on $\mathcal{C}$ and can be significantly better than the VC-dimension of the set of all (linear) classifiers in $\mathbb{R}^d$ (i.e., $d$)?

**Edit:** Above, $\boldsymbol{1}\hspace{-0.9ex}\mathrm{1}(\cdot)$ denotes the binary indicator function. So a more explicit definition of $\mathcal{H}$ is the set of functions $$f_\boldsymbol{h}(\boldsymbol{x})=\begin{cases} 0 & \text{if }\boldsymbol{h}^\mathsf{T}\boldsymbol{x}<0 \\ 1 & \text{if }\boldsymbol{h}^\mathsf{T}\boldsymbol{x}\ge0\end{cases},$$ parametrized by $\boldsymbol{h}\in \mathcal{C}$.

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