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Let $K>1$ be a positive integer. Consider a function class $\mathcal{F}_K:=\{\max_{1\leq k\leq K} a_k^\top x + b_k\}$ on some compact region, where $a_k\in\mathbb{R}^d$ and $b_k$ are bounded. Set $-\mathcal{F}_K:=\{-f:f\in\mathcal{F}_K\}$. What is the convex hull of $\mathcal{F}_K\cup\{-\mathcal{F}_K\}$?

Let $K>1$ be a positive integer. Consider a function class $$\mathcal{F}_K:=\Big\{\max_{1\leq k\leq K} a_k^\top x + b_k:\ ||a_k||\leq 1, |b_k|\leq 1, \forall 1\leq k\leq K \Big\}$$ on some compact region. Set $-\mathcal{F}_K:=\{-f:f\in\mathcal{F}_K\}$. What is the convex hull of $\mathcal{F}_K\cup\{-\mathcal{F}_K\}$?

Let $K>1$ be a positive integer. Consider a function class $\mathcal{F}:=\max_{1\leq k\leq K} a_k^\top x + b_k$ on some compact region, where $a_k\in\mathbb{R}^d$ and $b_k$ are bounded. Set $-\mathcal{F}:=\{-f:f\in\mathcal{F}\}$. What is the convex hull of $\mathcal{F}\cup\{-\mathcal{F}\}$?

Let $K>1$ be a positive integer. Consider a function class $\mathcal{F}_K:=\{\max_{1\leq k\leq K} a_k^\top x + b_k\}$ on some compact region, where $a_k\in\mathbb{R}^d$ and $b_k$ are bounded. Set $-\mathcal{F}_K:=\{-f:f\in\mathcal{F}_K\}$. What is the convex hull of $\mathcal{F}_K\cup\{-\mathcal{F}_K\}$?