Higher order concentration in presence of Poincar\'e-type inequalities

We show sharpened forms of the concentration of measure phenomenon typically centered at stochastic expansions of order $d-1$ for any $d\in \mathbb{N}$. Here we focus on differentiable functions on the Euclidean space in presence of a Poincar'e-type inequality. The bounds are based on $d$-th order derivatives.