SIMON-MARAIS 2023 A2

Let $n$ be a positive integer and let $f_1(x), \ldots, f_n(x)$ be affine functions from $\mathbb{R}$ to $\mathbb{R}$ such that, amongst the $n$ graphs of these functions, no two are parallel and no three are concurrent. Let $S$ be the set of all convex functions $g(x)$ from $\mathbb{R}$ to $\mathbb{R}$ such that for each $x \in \mathbb{R}$, there exists $i$ such that $g(x) = f_i(x)$.

Determine the largest and smallest possible values of $|S|$ in terms of $n$.

(A function $f(x)$ is affine if it is of the form $f(x) = ax + b$ for some $a, b \in \mathbb{R}$. A function $g(x)$ is convex if $g(\lambda x + (1 - \lambda)y) \leq \lambda g(x) + (1 - \lambda)g(y)$ for all $x, y \in \mathbb{R}$ and $0 \leq \lambda \leq 1$.)