Optimal bandwidth for a Gaussian filter

Question

I have an $n \times n$ image $A$, and an $m\times m$ image $B$, where $n>m$. As the smaller image looks like a lower-resolution version of the larger one, I'm interested in the relative loss, computed as follows:

• 1. use a Gaussian filter to low-pass $A$
• 2. downsample the filtered $A$ to the size of $B$
• 3. take the difference with $B$.

Specifically, let $f_\sigma$ be the Gaussian filter with bandwidth $\sigma$, and $\Phi(\cdot)$ be the function that downsamples every $n//m$ pixels along row and column. I want to compute $$\ell_\sigma = \lVert\Phi(f_\sigma \ast A) - B\rVert_F^2.$$

Here $\lVert\cdot\rVert_F$ is the Frobenius norm. $\ast$ is 2D convolution. I use subscript $\sigma$ because obviously the loss value depends on what bandwidth $\sigma$ we adopt.

I calculated the $\ell_\sigma$ for a range of $\sigma$'s. It seems $\ell_\sigma$ is a convex function of $\sigma$. Is that true for every $A$ and $B$? Is it possible to prove it?