As you move into higher dimensions, the phenomenon known as concentration of measure becomes increasingly important. This idea builds on the earlier discussion of how distances between points in high-dimensional spaces tend to collapse, meaning that the difference between the nearest and farthest neighbors shrinks. In practical terms, this leads to a surprising uniformity: most points in a large, high-dimensional dataset become nearly equidistant from each other. The space becomes so vast and the data so sparse that the extremes — those rare points that once stood out as outliers — almost vanish. Instead, almost all points look similar in terms of their geometric relationships.

This uniformity has deep consequences for how you interpret and analyze high-dimensional data. In low dimensions, you might expect to see some points that are clearly distinct or far away from the rest. In high dimensions, however, the concentration of measure means that such outliers are exceedingly rare. The bulk of the data is squeezed into a thin shell at a typical distance from the center, and the probability of finding a point that is much farther or much closer than this average is extremely small.