What is the nature of this one dimensional function?

Question

Let $\mathcal{S}$ be a 2-D convex set whose elements can be represented as $(x,y)\in\mathcal{S}$. Let $p_L$ and $p_U$ be two real constants such that $p_L\leq p_U$. For $p\in[p_L,p_U]$, I define the function \begin{align} g(p)=\min_{(x,y)\in\mathcal{S}}~& x,~~~\\ \mbox{such that} ~~&\frac{p_L}{p}\leq y \leq \frac{p_U}{p} \end{align}
Now for the same interval of $p$, define the function \begin{align} f(p)=p*g(p) \end{align} what is the nature of f(p)? (montonic increasing or decreaing, convex or concave and so on)