As you move into the realm of high-dimensional spaces, you encounter a phenomenon that seems counterintuitive at first: concentration of measure. This concept describes how, as the number of dimensions grows, the behavior of points in a space becomes surprisingly uniform. Earlier, you saw how the volume of high-dimensional shapes like spheres and cubes tends to concentrate near their surfaces, and how distances between random points collapse toward a common value. These geometric intuitions lay the groundwork for understanding why concentration of measure arises.

Imagine you are working in a space with hundreds or thousands of dimensions. If you randomly pick points from this space, you might expect them to be spread out in all sorts of ways. However, what actually happens is that almost all points end up being very similar with respect to many properties. For example, if you compute the distance from the center of a high-dimensional sphere to a random point, nearly all points will be at almost exactly the same distance. Likewise, if you measure a function (like the sum or average of coordinates) on these points, most of the values you get will cluster tightly around a single value — the mean.

This geometric uniformity is not just a quirk of one particular function or shape; it is a broad principle that applies to many different kinds of functions and high-dimensional spaces. The key intuition is that as the number of dimensions increases, the diversity of behaviors among random points actually decreases for many properties, making the "typical" point highly representative of the whole space.