Question: I am interested whether a concept for the multivariate equivalent to quantiles exists. 
In the univariate case, a $p$-quantile can be computed via the inverse of the cumulative density function, however, this cannot easily be translated into the multivariate setting. 
Whereas in the univariate setting we are able to denote the $\alpha$ quantile $p\in\mathbb{R}$ such that $F(p)=\alpha$. Given we have a $N$-variate distribution, isn't it possible to define an $N-1$ dimensional subspace (for example an ellipse) if we are talking about bivariate distribution) such that a similar statement can be made?

Answer: 
Answer to this question can be found [**in this post**](https://stats.stackexchange.com/questions/64680/how-to-determine-quantiles-isolines-of-a-multivariate-normal-distribution)  in the CrossValidated stackexchange forum.
i find it quite funny, but K-NN (k-nearest neighbors) and SOM (Kohonen's Self-organizing maps) and "vector quantization" are all about the same. They all give you "Multivariate Quantiles" (or "multidimensional quantiles").