Special Configurations of Planes Associated with Convex Compacta

Abstract

Several combinatorial geometry properties of convex compact sets are proved by topological methods. It is proved that if K 1,... ,K n-1 are convex compacta in $$\mathbb{R}^n $$ , then there is an (n-2)-plane $$E\subset\mathbb{R}^n $$ such that for each i=1,2,... ,n-1 there exist three (two orthogonal) hyperplanes containing E and dividing Ki into six (four) parts of equal volume. It is also proved that for every two bounded continuous distributions of masses in $$\mathbb{R}^3 $$ centrally symmetric with respect to the origin there are three planes dividing both masses into eight equal parts. Bibliography: 9 titles.