Smoothness parameter of power of Euclidean norm

Abstract

In this paper, we study derivatives of powers of Euclidean norm. We prove their H\"older continuity and establish explicit expressions for the corresponding constants. We show that these constants are optimal for odd derivatives and at most two times suboptimal for the even ones. In the particular case of integer powers, when the H\"older continuity transforms into the Lipschitz continuity, we improve this result and obtain the optimal constants.