  Busemann's theorem states that the intersection body of an origin-symmetric
convex body is also convex. In this paper we provide a version of Busemann's
theorem for p-convex bodies. We show that the intersection body of a p-convex
body is q-convex for certain q. Furthermore, we discuss the sharpness of the
previous result by constructing an appropriate example. This example is also
used to show that IK, the intersection body of K, can be much farther away from
the Euclidean ball than K. Finally, we extend these theorems to some general
measure spaces with log-concave and $s$-concave measures
