Poincaré and Logarithmic Sobolev Inequalities for Nearly Radial Measures

Poincaré inequality has been studied by Bobkov for radial measures, but few are known about the logarithmic Sobolev inequality in the radial case. We try to fill this gap here using different methods: Bobkov’s argument and super-Poincaré inequalities, direct approach via L1-logarithmic Sobolev inequalities. We also give various examples where the obtained bounds are quite sharp. Recent bounds obtained by Lee–Vempala in the log-concave bounded case are refined for radial measures.