What are the multidimensional versions of the median and what are their pros and cons? I confess this doesn't have a single answer, but I think it is a useful question to ask and will be a benefit to others as well.
How stable it is (i.e. how many samples one needs to get a reasonable estimate of it) is one potential, but not necessary, pro and con issue, i.e. if you know that the number of samples necessary grows exponentially with each increase in dimension making it useful in 10 dimensions but effectively useless in 200 dimensions, that would be useful to know. (I would kind of expect all of them have that, really, because there is so much "freedom" in 200 dimensions it just takes a bazillion points to nail the basic middle of a distribution down along that many directions).
Note: I found this question after accepting an answer for this. I hadn't seen it before because I didn't realize there was both a multivariate-analysis and a multivariable tag. I am not sure why geometric median was an unpopular answer there (1 vote and near the bottom) but a popular answer here.