Isoperimetric inequalities for certain star-shaped bodies

I will present isoperimetric type inequalities for star-shaped sets that interpolate between intersection bodies and dual Lp centroid bodies, two classes of convex bodies studied in geometry. This provides a bridge between two well-known inequalities: the Busemann intersection inequality and the Lutwak-Zhang inequality. I will also describe inequalities for random counterparts of these sets. Based on joint work with Grigoris Paouris, Peter Pivovarov and Paul Simanjuntak.