*Note: This is a follow-up question inspired by a previous (more difficult) question I asked on MathOverflow.*

Let $f:\mathbb{R}\to\mathbb{R}$ be a (sufficiently regular, e.g. smooth) probability density supported on the (compact) interval $[a,b]$. Let $\Vert\cdot\Vert$ be some norm (e.g. $L^2$, total variation). What is the best approximation to $f$ by a single, univariate Gaussian? In other words, solve $$ \text{argmin}_{m\in\mathbb{R},v^2\ge 0}\Vert f-g_{m,v^2}\Vert $$

where $g_{m,v^2}$ is the Gaussian PDF with mean $m$ and variance $v^2$. As I mention above, this question is related to a previous question; I realized while trying to solve that problem that even the simple case of a single Gaussian seems nontrivial.

**Hint:** I have verified numerically, for both $L^2$ and TV norm, that **the answer is not $m=\mathbb{E}_{X\sim f} X$ or $v^2=\text{var}_{X\sim f} (X)$**.

**Note:** The details of which norm (or even metric) or regularity assumptions are made are not important here. I am mostly curious to see if this computation can be carried out for any "reasonable" distance and regularity. In fact, even the case of minimizing over $m$ alone (holding $v^2$ fixed) appears difficult. Note that for the KL-divergence (which is neither a norm nor a metric), this can be solved in closed form quite easily.