A note on Poincaré constants

Abstract

In this note, we give two comparison results of Poincar\'{e} constants:

(i) Let $X$ and $Y$ be Banach spaces. We give a relationship between the $X$-valued $p$-Poincar\'{e} constant and the $Y$-valued $q$-Poincar\'{e} constant introduced by Laat and Salle for all $1\leq p,q<\infty$ when the unit sphere $S(X)$ of $X$ is uniformly homeomorphic to the unit sphere $S(Y)$ of $Y$.

(ii) We provide an explicit relationship between the nonlinear spectral gap introduced by Mimura and the Poincar\'{e} constant introduced by Laat and Salle.