# Convex hull of piece-wise linear functions

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 $\mathcal{F}_K\cup\{-\mathcal{F}_K\}$?

• Since $\mathcal{F}$ is a convex cone, the convex hull is the same set as $\mathcal{F}-\mathcal{F}$ – Pietro Majer Aug 13 '18 at 16:28
• Isn't the convex hull of $\mathcal{F}$ the set of all convex functions? – Dirk Aug 13 '18 at 16:57
• @PietroMajer Why is $\mathcal{F}$ convex? The number of pieces is fixed as $K$. – O. Richard Aug 13 '18 at 17:32
• @Dirk Is it obvious? Here the number of pieces is fixed. – O. Richard Aug 13 '18 at 17:33
• But taking the convex hull should allow at least countably many kinks. Or are these $a_k$'s and $b_k$'s fixed? – Dirk Aug 13 '18 at 17:42