  Probability measures satisfying a Poincar{\'e} inequality are known to enjoy
a dimension free concentration inequality with exponential rate. A celebrated
result of Bobkov and Ledoux shows that a Poincar{\'e} inequality automatically
implies a modified logarithmic Sobolev inequality. As a consequence the
Poincar{\'e} inequality ensures a stronger dimension free concentration
property , known as two-level concentration. We show that a similar phenomenon
occurs for the Latala-Oleszkiewicz inequalities, which were devised to uncover
dimension free concentration with rate between exponential and Gaussian.
Motivated by the search for counterexamples to related questions, we also
develop analytic techniques to study functional inequalities for probability
measures on the line with wild potentials.
