You missed it. Bourgain guessed that some of these lower-dimensional slices must have substantial area. In particular, he conjectured that there is some univer

You missed it. Bourgain guessed that some of these lower-dimensional slices must have substantial area. In particular, he conjectured that there is some universal constant, independent of the dimension, such that every shape contains at least one slice with area greater than this constant. In other words there exists C > 0 such that if the n-dimensional hypervolume of an n-dimensional convex shape is 1, then there must be an n-1'th dimensional slice of n-1-dimensional hypervolume at least C. What was proven is weaker. For any ε > 0 there is an N such that if N < n, then any n-dimensional convex shape of n-dimensional hypervolume 1 must have an n-1'th dimensional slice of n-1-dimensional hypervolume at least 1/n^ε. It turns out that the exact things that were proven are good enough to improve our bounds on how quickly various machine learning algorithms will converge. Which means we aren't just hoping based on how they worked in a few examples, we have a theory explaining it.