In this note, we give two comparison results of Poincaré constants: (i) Let X and Y be Banach spaces. We give a relationship between the X-valued p-Poincaré constant and the Y-valued q-Poincaré constant introduced by Laat and Salle for all 1 ≤ p, q < ∞ 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 Poincaré constant introduced by Laat and Salle.