The hyperplane problem asks whether there exists an absolute constant such that every symmetric convex body of volume one in every dimension has a central hyperplane section with area greater than this constant. We consider a generalization of this problem to arbitrary measures in place of volume and to sections of lower dimensions. We prove this generalization for unconditional convex bodies, for duals of bodies with bounded volume ratio, and for k-intersection bodies. We also prove it for arbitrary symmetric convex bodies with a constant dependent only on the dimension, and with an absolute constant when sections are of proportional dimension.
