It's an example of concentration of measure: https://en.wikipedia.org/wiki/Concentration_of_measure

It's easy to verify this for yourself in a simple example in something like python. Draw from a multivariate normal distribution (or almost anything) in a 1000-dimensional space and look at the norm of your vector. If you do this a lot of times you'll see that the answer is close to the same each time. This even holds true if you replace the vector norm by the distance to an arbitrary fixed point.

edit: here's some code:

import numpy import matplotlib.pyplot norms = [] dists = [] for i in range(5000): vec = numpy.random.normal(size=4096) norms.append(numpy.linalg.norm(vec)) dists.append(numpy.linalg.norm(vec-numpy.ones(4096))) matplotlib.pyplot.hist(norms) matplotlib.pyplot.show() matplotlib.pyplot.hist(dists) matplotlib.pyplot.show()

(didn't render right, sorry)