  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.
