Mann–Whitney $U$test in $n$ dimensions This https://math.stackexchange.com/q/1576971/290307 question reminded me of a yet unanswered question I had as a student.
https://en.wikipedia.org/wiki/Mann%E2%80%93Whitney_U_test allows me to decide if two samples in $\mathbb R$ have significantly different distributions.
But for samples in $\mathbb R^n$ I have a choice of normal vectors against which I could order the sample sets. How would I compensate for that?
EDIT:
To clarify, my question is not about more than two samples but about the sample point being in $\mathbb R^n$ and thus lack a canonical ranking.
By picking a hyperplane they get a ranking as the signed distance from that plane and I could do the MW-test.
But as there are many hyperplanes I could pick and test against, there is an increasing chance that at least one of the choices gives a false positive.
 A: The problem of testing for the equality of $g$ group means is a well-discussed in nonparametric inference. Here are brief mentions of three methods in common use.
Solution (1) would be to look into the theory behind the Kruskal-Wallis test, which deals with exactly this issue. (K-W for two groups is precisely M-W.) Perhaps start with the brutally brief account in Wikipedia on 'Kruskal-Wallis test', and then graduate to the references there. 
As with all rank-based tests, the K-W test is
intended for data from continuous populations, and so becomes
problematic if there are many ties in the data (within or among
samples).
(2) would be to do ${n \choose 2}$ M-W tests comparing two samples at a time, using level $\alpha/n$ for a 'family' error rate not exceeding $\alpha$ (according to Bonferroni's Inequality). Again here, ties can lead to difficulties.
(3) Would be to do a permutation test, perhaps using the F-statistic of a one-way ANOVA as metric; a simulated permutation distribution would approximate the actual distribution of the F-statistic, which might not be an F-distribution.
If you have a reasonably brief dataset with random samples from 3 or 4 populations, please edit it into your question and I will try to provide analyses using as many of these methods as are of interest to you. I do not think this is an appropriate forum for showing theoretical details exposited elsewhere. (Otherwise, I can provide an appropriate dataset for demonstration.) 
