Stability of the $L^{p}$-Poincaré inequality for the Lebesgue measure and Gaussian probability measure with explicit geometric dependence and applications to spectral gaps

In this paper, we obtain stability results for the $L^{p}$-Poincaré inequality for both Lebesgue measure and Gaussian probability measure (Theorem 3.3 and Theorem 3.13) that involve explicit dependence on the geometry of the domain. As a byproduct, the explicit constant allows us to recover importan