**L1 distance between gaussian measures: Definition**


Let $P\_1$ and $P\_0$ be two gaussian measures on $\mathbb{R}^p$ with respective "mean,Variance" $m\_1,C\_1$ and $m\_0,C\_0$ (I assume matrices have full rank). I know that calculating the L1 distance between $P\_1$ and $P\_0$: 
$$d\_1=\int|dP\_1-dP\_0|$$


**The easy case**


is an easy exercice when $C\_0=C\_1=C$: 
$$d\_1=2(2\Phi(\sigma/2)-1)$$
where
$$\sigma=\|C^{-1/2}(m\_1-m\_0)\|.$$
(norm of $\mathbb{R}^p$) and $\Phi$ is the cdf of a gaussian mean zero variance 1 reall variable.


I don't remember the name of $\sigma$ (RKHS norm? Cameron martin ?) but it can also be written: $\|\mathcal{L}\|\_{L\_2(P\_{1/2})}$ where $\mathcal{L}$ is the log likelihood ratio function and $P\_{1/2}$ is the normal distribution with mean $(m\_1+m\_0)/2$ and variance $C$. 


**My question is about how to extend that type of result for the case when $C\_0\neq C\_1$** (explicit calculation of the L1 distance).


I see two possible reductions of the problem if calculous are too complicated:


1. search for an inequality relating the L1 distance and some norm of the likelihood ratio
2. search for some exact expression in a particular case, for example $C\_1$ and $C\_0$ diagonal.


Reduction 1 gets a partial answer with the general inequality


$$d\_1\leq 2\sqrt{K(P\_1,P\_0)}$$
(due to pinsker or Lecam I don't remember)
where 
$$K(P\_1,P\_0)=\int \log \left(\frac{dP\_1}{dP\_0} \right ) dP\_1$$
is the kullback divergence. 


I am not really satisfyed with this answer since in the case $C\_1=C\_0$ it is suboptimal, it does not include an "half measure" $P\_{1/2}$ (could include $(P\_0+P\_1)/2$ by using twice the inequality but I don't really like this interpolation),... 



L1 distance between gaussian measures
Explicit upper and lower bounds are obtained in Theorem 1.2 and Proposition 2.1 of [The total variation distance between
high-dimensional Gaussians](https://arxiv.org/pdf/1810.08693.pdf).


