Comparing the Relative Volume with the Relative Inradius and the Relative Width

We consider subdivisions of a convex bodyG in two subsets E andG \E. We obtain several inequalities comparing the relative volume: (1) with the minimum relative inradius, (2) with the maximum relative inradius, (3) with the minimum relative width, and (4) with the maximum relative width. In each case, we obtain the best upper and lower estimates for subdivisions determined by general hypersurfaces and by hyperplanes.