Quantitative form of certain k-plane transform inequalities

Let d≥2d≥2 and 1≤k≤d−11≤k≤d−1. The k  -plane transform satisfies some Lp→LqLp→Lq dilation-invariant inequality. In this case the best constant and the extremizers are explicitly known. We give a quantitative form of the inequality with respect to these extremizers, that works for k=d−1k=d−1 and for k≠d−1k≠d−1 while restricted to radial functions.