It is well known that for one dimensional random variables $$\cases{pdf(t) : 1=\int_{-\infty}^{\infty}pdf(t)dt\\cdf(t) = \int_{-\infty}^{t}pdf(\tau)d\tau}$$ The k:th percentile is the solution $t_k$ for which: $$cdf(t_k)=\frac k {100}$$ Where the well known special case the median, $k=50$.
However for multivariate distributions, how can one define these things uniquely without ending up with manifolds that are dependent on the coordinate system?
If it is not possible to do, is there still any use ( practical applications ) to this generalization which results in larger-than-0-dimensional level sets?