Let $T$ be a compact, connected, proper subset of $\mathbb{R}^3:\quad T \subset \mathbb{R}^3$.

Further let $\left\{ \boldsymbol{\mu}_i \right\}_{i=1}^n$ be a given finite set of $n$ point in $T$: $$ \left\{ \boldsymbol{\mu}_i \right\}_{i=1}^n \subset T $$

I would like to approximate continuous uniform distribution $u(\boldsymbol{x})$ over $T$ with mixture of Gaussians $g(\boldsymbol{x})$, for which means of components are given by the set $\left\{ \boldsymbol{\mu}_i \right\}_{i=1}^n$, that is: $$ g(\boldsymbol{x}) = \sum_{i=1}^{n} \pi_{i} \mathcal{N}(\boldsymbol{x} \,|\, \boldsymbol{\mu}_{i},\Sigma_{i}) $$

We can assume that, informally speaking, means $\left\{ \boldsymbol{\mu}_i \right\}_{i=1}^n$ are "equispaced" over $T$ or "nice" enough such that good approximation exists.

Any ideas on this?

The way I want to attack this is by minimizing some sort of computationally tractable divergence (Kullback–Leibler divergence, for example) between $u$ and $g$.

I'm curious if something similar was attempted before (searching the web didn't yield anything)?

Or could you point me to the relevant literature on this? Basically, any suggestions would be of help.

**Motivation**: this problem arises as an inverse problem in radiotherapy treatment planning. $T$ is considered to be a tumor and needs to receive radiation dose that is uniformly distributed over $T$. Dose spillage outside $T$ is undesirable. In this case the problem of designing such treatment can be reduced to estimating values of weights $\; \pi_1,\, \dots,\, \pi_n\;$ and covariance matrices $\; \Sigma_1,\, \dots,\, \Sigma_n$.