# pr.probability – Relationship between \$L_1\$ and \$L_2\$ distances of two Gaussian Mixture models

Given two Gaussian mixture models with
begin{equation} begin{aligned} f(x) &=sum_{k=1}^{K} pi_{k} mathcal{N}left(x mid mu_{k}, sigma_{k}right), \ g(x) &=sum_{i=1}^{N} lambda_{i} mathcal{N}left(x mid mu_{i}, sigma_{i}right), end{aligned} end{equation}
I want to compute their $$L_1$$ distance,
$$begin{equation} delta_{1}(f,g) = int_{x} |f(x) – g(x)|. end{equation}$$
However, the $$L_1$$ distance cannot be computed directly, and I use the squared $$L_2$$ distance instead, i.e.,
$$begin{equation} delta_{2}(f,g) = int_{x} (f(x) – g(x))^2. end{equation}$$
The closed-form expression of $$L_2$$ distance is available (c.f. Distance between two Gaussian mixtures to evaluate cluster solutions), but I want to know: does there exist any relationship between $$L_1$$ distance and $$L_2$$ distance in this case?