Median of 2 Dimensional Vectors

We had a discussion in class earlier today about whether the median of a set should actually be an element from the set in question itself; and if this were the case, the method described in class of taking the median of 2 vectors might not necessarily hold true because taking the median of the x components and the median of the y components and combining them might result in a vector not originally present in the set.

However, it turns out that even for 1-dimensional data, whenever we have an even number of elements in the set, the median always turns out to be the mean of the 2 middle elements (when the set is arranged in non-increasing or non-decreasing order).

Therefore the idea put forward in class would seem reasonable, that even if the "median vector" itself is not part of the set, it can be thought of as the median.