The lower dimensional Busemann-Petty problem asks whether origin symmetric convex bodies in $\mathbb{ R}^n$ 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 $\mathbb{ H}^n$.
Categories: math.FA math.MG