  Busemann's intersection inequality gives an upper bound for the volume of the
intersection body of a star body in terms of the volume of the body itself.
Koldobsky, Paouris, and Zymonopoulou asked if there is a similar result for
$k$-intersection bodies. We solve this problem for star bodies that are close
to the Euclidean ball in the Banach-Mazur distance. We also improve a bound
obtained by Koldobsky, Paouris, and Zymonopoulou for general star bodies in the
case when $k$ is proportional to the dimension.
