# Smoothness parameter of power of Euclidean norm

## TL;DR

This paper investigates the derivatives of Euclidean norm powers, proving their Hölder continuity, deriving explicit constants, and optimizing these constants for odd derivatives and integer powers.

## Contribution

It provides explicit expressions and optimal constants for the Hölder continuity of derivatives of Euclidean norm powers, improving understanding of their smoothness properties.

## Key findings

- Constants are optimal for odd derivatives.
- Constants are at most twice suboptimal for even derivatives.
- Improved results for integer powers with Lipschitz continuity.

## 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.