ε-Entropy and moduli of smoothness in ${\displaystyle L^p}$-spaces

The asymptotic behaviour of ε-entropy of classes of Lipschitz functions in ${\displaystyle L^p(^d)}$ is obtained. Moreover, the asymptotics of ε-entropy of classes of Lipschitz functions in ${\displaystyle L^p\left({ℝ}^d\right)}$ whose tail function decreases as ${\displaystyle O\left({\lambda }^{-\gamma }\right)}$ is obtained. In case p = 1 the relation between the ε-entropy of a given class of probability densities on ${\displaystyle {ℝ}^d}$ and the minimax risk for that class is discussed.