I am looking for the volume of the following ensemble, which is the union of $K$ ellipsoids:
$$ \left\{ x\in\mathbb R^d : \sum_{k=1}^K \frac{\exp\left[- (x-m_k)^T \Sigma_k^{-1} (x-m_k)/2 \right]}{\sqrt{\det \Sigma_k}} \leq c\right\}, $$
where $K, d \in \mathbb N$, $m_k \in \mathbb R^d$, $\Sigma_k \in \mathbb R^{d \times d}$ et $c>0$.
For $K=1$, the answer is obviously $\frac{\pi^{d/2}}{\Gamma(d/2+1)}\sqrt{\det\left( -\Sigma_1\ln(c^2 \det \Sigma_1) \right)}$, but I cannot find the answer for $K>1$.
Thank you for your help.