3 ms·
Did they define "substantial area" and I just missed it?
by mcguire 6y ago
Did they define "substantial area" and I just missed it?
- mbeex 6y agoNo, and in particular by speaking about slices with a lower dimension d-1 it becomes not immediately clear, what their volume has to do with the one in dimension d of the original body, mentioned at the begin of the article.
- btilly 6y agoYou 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.