Sharp Quantitative Stability for the Affine $p$-Sobolev Inequality, Part II: The Case $1<p<2$

We prove a sharp quantitative stability result for the affine $L^p$-Sobolev inequality, for $1<p<2$, introduced by Lutwak–Yang–Zhang (\emph{J. Differential Geom.}, $\mathbf{\text{62}}$ (2002),17–38). Moreover, the stability exponent is shown to be optimal, and equal to $2$.