Equivalence of concentration inequalities for linear and non-linear functions

We consider a random variable $X$ that takes values in a (possibly infinite-dimensional) topological vector space $\mathcal{X}$. We show that, with respect to an appropriate "normal distance" on $\mathcal{X}$, concentration inequalities for linear and non-linear functions of $X$ are equivalent. This normal distance corresponds naturally to the concentration rate in classical concentration results such as Gaussian concentration and concentration on the Euclidean and Hamming cubes. Under suitable assumptions on the roundness of the sets of interest, the concentration inequalities so obtained are asymptotically optimal in the high-dimensional limit.