Mean distance between points inside and on the boundary of a convex body

The Zaporozhets--Tarasov conjecture is considered, which states that the mean distance between two random points inside a convex body does not exceed the mean distance between two random points on its boundary. For centrally symmetric bodies the question is settled completely: the conjecture is proved for planar bodies and disproved in every dimension at least three, for all moments of the distance; an explicit family of counterexamples is constructed together with a dimension-lifting construction. It is also shown that for sufficiently high moments an analogous inequality holds for any planar convex body. For circumscribed bodies exact relations between the mean distances are obtained --- analogues of Kingman's formula derived via the projective distributions introduced here; moreover, the conjecture is proved for triangles in a stronger per-projection form.