On sections of convex bodies in hyperbolic space

We consider two problems on sections of convex bodies in hyperbolic space. The first one is a modified version of the Busemann–Petty problem. We look at conditions that guarantee a positive answer to this problem in all dimensions. The second problem is an analogue of a result of Makai, Martini, and Ódor about origin-symmetry. If in every direction the parallel section function has a critical value at zero, then the body is origin-symmetric. For both problems we use Fourier transform techniques.