Given a compact closed convex set $C \subset \mathbb{R}^n$, its known that for a given $y \in \mathbb{R}^n$, the following is considered a linear program: $$ \max_{x \in C} x \cdot y, $$ where the $\cdot$ represents the dot product.
Instead, we set our objective to be: $$ \max_{-1 \preceq y \preceq 1} \max_{x \in C} x \cdot y, $$ where $-1 \preceq y \preceq 1$ means that every entry of $y$ is between $1$ and $-1$.
Is this problem considered to be linear program? And if not, what is it's type? Can it be solved using any techniques from linear/quadratic/convex optimization?
Please advise and thanks in advance.
As I understand your question, your problem can be re-stated as
\begin{equation*} \begin{array}{lllr} \max & xy\\ \textrm{subject to } & x &\in &C \\ & y &\leq &1 \\ & y &\geq &-1 \end{array} \end{equation*}
You are turning your entries $y$ into variables that assumes real numbers between -1 and 1. This can be viewed as a non-linear program since its objective function has a product of two variables. See https://en.wikipedia.org/wiki/Nonlinear_programming#Definition
However, if you take a closer look, you can see that there always exists an optimal solution where the values of $y$ are restricted to either 1 or -1 (if $x \geq 0$ then $y=1$, if $x < 0$ then $y=-1$).
So your problem can be rewritten as \begin{equation*} \begin{array}{lllr} \max & \left| \left| x \right| \right|_1\\ \textrm{subject to } & x &\in &C \end{array} \end{equation*}
which can be easily linearized in order to find a linear program that models your problem.