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We establish existence of Stein kernels for probability measures on$\mathbb{R}^d$ satisfying a Poincar\'e inequality, and obtain bounds on theStein discrepancy of such measures. Applications to quantitative central limittheorems are discussed, including a new CLT in Wasserstein distance $W_2$ withoptimal rate and dependence on the dimension. As a byproduct, we obtain astability version of an estimate of the Poincar\'e constant of probabilitymeasures under a second moment constraint. The results extend more generally tothe setting of converse weighted Poincar\'e inequalities. The proof is based onsimple arguments of calculus of variations.  Further, we establish two general properties enjoyed by the Steindiscrepancy, holding whenever a Stein kernel exists: Stein discrepancy isstrictly decreasing along the CLT, and it controls the skewness of a randomvector.

Existence of Stein Kernels under a Spectral Gap, and Discrepancy Bound