Geometric asymptotics and the logarithmic Sobolev inequality

The logarithmic Sobolev inequality is developed as a geometric asymptotic estimate With respect to Lebesgue measure. Two short geometric arguments are given to derive (1) the logarithmic Sobolev inequality from the isoperimetric inequality and (2) Nash’s inequality With an asymptotically sharp constant from the logarithmic Sobolev inequality. In addition, the Fisher information form of the logarithmic Sobolev inequality is obtained directly from the isoperimetric inequality. A new formulation of the logarithmic Sobolev inequality is given for hyperbolic space which can be interpreted as an “uncertainty principle” in this setting.