Special configurations of planes associated with convex compact

Topological methods are applied for proving several combinatorial geometry properties of convex compact sets. It is proved that if $K_1,\dots,K_{n-1}$ are convex compacta in $\mathbb R^n$, then there is an $(n-2)$-plane $E\subset\mathbb R$ such that for $i=1,2,\dots,n-1$ there exist three (two orthogonal) hyperplanes through $E$ dividing each of $K_i$ into six (four) parts of equal volume. It is also proved that for every two bounded centrally symmetric continuous distributions of masses in $R^3$ with common center of symmetry there are three planes through this center, dividing both masses into eight equal parts.