Is the following function a convex function?

Question

$X\in \mathbb{R}^{m\times n}$ is a matrix variable and $A\in\mathbb{R}^{p\times m}$ is a constant matrix. Is the following function is a convex function with respect to $X$: $$f(X) = \|\max_{j}AX-\min_{j}AX\|_{2},$$ where $\max_{j}$ results in a column vector, each element of which is the maximum of the corresponding row, $\min_{j}$ also results in a column vector, each element of which is the minimum of the corresponding row, $\|\cdot\|_{2}$ denotes the Euclidean norm of a vector.