In this remarkable paper 30 pages are occupied by the proof of the following innocently looking lemma:

Let $K$ be an origin-symmetric convex body in $\mathbb R^3$. There exist three planes through the origin splitting $K$ into $8$ parts of equal volume and such that each two of these planes split the cross-section of $K$ by the third one into $4$ parts of equal area.

I cannot shake off the feeling that there must be a half-page proof of this statement though I don't have one yet. I also know that MO is swarming with good topologists. Anybody up to the challenge?

depend on the planes. However the idea that if the solution is unique insomegeneric position, then there is an odd number of solutions ineverygeneric position is amazing and certainly very promising. It should be rather standard, of course, but the good side of ignorance (mine) is the possibility to get surprised with the facts everybody else considers routine :-). $\endgroup$ – fedja Jun 12 '17 at 15:30