Isomorphic Busemann–Petty problem for sections of proportional dimensions

The main result of this note is a solution to the isomorphic Busemann–Petty problem for sections of proportional dimensions, as follows. Suppose that 0<λ<10<λ<1, k>λnk>λn, and K,LK,L are origin-symmetric convex bodies in RnRn satisfying the inequalities|K∩H|≤|L∩H|,∀H∈Grn−k, where Grn−kGrn−k is the Grassmanian of (n−k)(n−k)-dimensional subspaces of RnRn, and |K||K| stands for volume of proper dimension. Then|K|n−kn≤Ck((1−log⁡λ)3λ)k|L|n−kn, where C is an absolute constant.