From Blaschke--Santaló-type inequalities to uniform contractions

In this short note, we establish Blaschke–Santaló-type inequalities for $r$-ball bodies. Building on these inequalities, we somewhat further extend earlier results on analogues of the Kneser–Poulsen conjecture concerning intersections of balls under uniform contractions in Euclidean $d$-space. As an immediate corollary, we obtain a proof of Alexander’s conjecture for uniform contractions.