One way to phrase the "concentration-of-measure" phenomenon is that, for a Euclidean sphere $S^d$ in $d$ dimensions, for large $d$, "most of the mass is close to the equator, for any equator."1
Q. How could one explain/justify this intuitively—perhaps just verbally—to a mathematically literate but naive audience (say, advanced undergraduate math majors)?
That "most of the mass is close to the equator, for any equator" seems almost contradictory (imagining orthogonal equatorial hyperplanes), or at the least, superficially quite puzzling. Can one only gain intuition via working through details of the Brunn–Minkowski theorem or the isoperimetric inequality?
1Boáz Klartag, in a book review in the AMS Bulletin, July 2015, p.540. According to the Wikipedia article, the idea goes back to Paul Lévy.