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 $\mathcal{F}_K\cup\{\mathcal{F}_K\}$?
$\begingroup$
$\endgroup$

$\begingroup$ Since $\mathcal{F}$ is a convex cone, the convex hull is the same set as $\mathcal{F}\mathcal{F}$ $\endgroup$ – Pietro Majer Aug 13 '18 at 16:28

$\begingroup$ Isn't the convex hull of $\mathcal{F}$ the set of all convex functions? $\endgroup$ – Dirk Aug 13 '18 at 16:57

$\begingroup$ @PietroMajer Why is $\mathcal{F}$ convex? The number of pieces is fixed as $K$. $\endgroup$ – O. Richard Aug 13 '18 at 17:32

$\begingroup$ @Dirk Is it obvious? Here the number of pieces is fixed. $\endgroup$ – O. Richard Aug 13 '18 at 17:33

$\begingroup$ But taking the convex hull should allow at least countably many kinks. Or are these $a_k$'s and $b_k$'s fixed? $\endgroup$ – Dirk Aug 13 '18 at 17:42