Why is this method not Robust to Outlier in Multivariate Median Calculation

Question

I have been going through 2 posts:

1. What are the multidimensional versions of median here

And

2. Is there an accepted definition for the median of a sample on the plane, or higher ordered spaces? here

In the first post @Bitwise mentions a method for calculating the median for multivariate data by finding the minimum Euclidean distance (assuming Euclidean distance can be used) from one point to all other points in the data, the point with the least distance can be considered to be the median (though I understand that multivariate median has many variations).

In the second post, @user28 pretty much mentions the same thing, however, as per the comment by @user603, the method does not seem to be robust to outliers.

I am a bit confused, if Euclidean distance is used then, why is the metric not robust to outliers? In case there are outliers (observably extreme observations on all dimensions) would the Euclidean distance not be greater in this case from each point to the outlier(s) and be added to the overall sum of distances for each point, thus making the comparison between the sum of distances, more or less, same as the sum of distances without the outlier(s)?

Answer 1

The change in total distance for points closer to the outliers would be lesser than the ones farther. In other words, the total-distance distribution would see its minima shift a little to the side of outliers, implying change in perceived median.

For a one-dimensional example, lets say we have outliers for $x$ on right extremes => the additional distance would look like a straight line with negative slope (say, $y = m - x$). Lets say your total-distance(without outliers) looks like $y = k + x^2$, now you add $y = m - x$ (additional distances due to outliers) to it $=>$ the minima for total-distance shifts a little to right