[krnsll]

Something I worked on in my PhD was analyzing high dimensional bodies via their "sections."

Here's the Busemann Petty Problem:

Given two origin symmetric convex bodies K and L in n dimensions. Suppose for every linear hyperplane A (passing through the origin) we have vol_{n-1}(K intersect A) \leq vol_{n-1}(L intersect A).

Is it true that vol_{n}(K) < vol_{n}(L)?

[Here vol_k should be thought of as length when k = 1, area when k = 2, and volume in the traditional sense in k = 3.... generalizes quite well to arbitrary dimensions. And sections are these quantities L (resp. K) intersect A]

Turns out the answer is NO! In n \geq 10, it can be explained with the simple examples of K and L being the unit volume (vol_n) cube and a euclidean ball of volume (vol_n) slightly less 1 respectively. Comes from Keith Ball who, in his PhD thesis, established that {n-1}-section volume of the unit volume cube lies in [1, \sqrt(2)]. However for the euclidean ball of unit volume the section volume is at least sqrt(2). So you can start with the unit volume ball, decrease its radius infinitesimally so (the n-1 section volume falls less than the n-volume does) and generate a clear counterexample.

What this looks like is a ball with volume less than a cube but section volume seemingly leaks out of the faces of the cube. So a "spikey ball," if you may.

[selimthegrim]

Does Brunn-Minkowski get used here at all wrt (maybe mixed) volume?

[krnsll]

Not at all, the proof was a very elegant argument involving fourier transforms and an integral estimate going back to the study of controlling random walks (Khintchine's inequality). I say elegant in the manner of it being enviably so -- a proof a beginning graduate student could follow while capturing a fundamental, easy to state fact.

This work does however situate itself in/adjacent to that broad space of Brunn-Minkowski theory.