Sharp Quantitative Stability for the Affine \(p\)-Sobolev Inequality, Part I: The Case \(2\le p<n\)

Abstract: We prove a sharp quantitative stability result for the affine (L^p)-Sobolev inequality, for (p\ge2), introduced by Lutwak--Yang--Zhang (\emph{J. Differential Geom.}, \textbf{62} (2002), 17--38). Moreover, the stability exponent is shown to be optimal, and equal to (p).