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

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 \(p\).