In 2019, A. S. Tarasov and D. N. Zaporozhets conjectured that the mean distance between
two random points on the boundary of a convex body is never smaller than the mean distance
between two random points inside it. In this talk, we will completely settle this conjecture
for centrally symmetric planar convex bodies. We will present exact relations between the
mean distances for these figures, which are analogues of Kingman’s formula linking the mean
distance between interior points to the length of a random chord. Finally, we will discuss
possible ways to relax the original conjecture for arbitrary planar bodies