Stability estimate for the Lane-Emden inequality

The Lane-Emden inequality controls ${\iint }_{{\mathbb{R}}^{2d}}\rho (x)\rho (y)\mathrm{\mid }x-y{\mathrm{\mid }}^{-\lambda }dxdy$ in terms of the $L^1$ and $L^p$ norms of $\rho$. We provide a remainder estimate for this inequality in terms of a suitable distance of $\rho$ to the manifold of optimizers.