Low-Temperature Asymptotics of the Poincaré and the log-Sobolev Constants for Łojasiewicz Potentials

In this paper, we establish the low-temperature asymptotics of the Poincaré inequality constant for a class of convex potentials satisfying a Łojasiewicz inequality. In addition, we disprove a conjecture previously posed by Chewi and Stromme on the low-temperature asymptotics of the log-Sobolev constant and determine the correct asymptotic behavior in dimension one.