The lower dimensional Busemann–Petty problem in the complex hyperbolic space

The lower dimensional Busemann–Petty problem asks whether origin-symmetric convex bodies in RnRn with smaller volume of all k  -dimensional sections necessarily have smaller volume. The answer is negative for k>3k>3. The problem is still open for k=2,3k=2,3. We study this problem in the complex hyperbolic n  -space HCn and prove that the answer is affirmative only for sections of complex dimension one and negative for sections of higher dimensions.