The Busemann theorem for complex p-convex bodies

Abstract

The Busemann theorem states that the intersection body of an origin-symmetric convex body is also convex. In this paper, we prove a version of the Busemann theorem for complex p-convex bodies. Namely that the complex intersection body of an origin-symmetric complex p-convex body is γ-convex for certain γ. The result is the complex analogue of the work of Kim, Yaskin, and Zvavitch on (real) p-convex bodies. Furthermore, we show that the generalized radial qth mean body of a p-convex body is γ-convex for certain γ.