A solution to the lower dimensional Busemann-Petty problem in the hyperbolic space

The lower dimensional Busemann-Petty problem asks whether origin symmetric convex bodies in ℝⁿ with smaller volume of all k-dimensional sections necessarily have smaller volume. As proved by Bourgain and Zhang, the answer to this question is negative if k>3. The problem is still open for k = 2, 3. In this article we formulate and completely solve the lower dimensional Busemann-Petty problem in the hyperbolic space ℍⁿ.