Symmetric norms and the Leibniz property

TL;DR

This paper demonstrates that specific symmetric seminorms on real vector spaces satisfy the Leibniz inequality, extending known results for standard deviation to a broader class of norms and connecting to non-commutative geometry.

Contribution

It establishes that certain symmetric seminorms on  satisfy the Leibniz inequality and links these results to non-commutative Riemannian metrics.

Findings

Symmetric seminorms satisfy Leibniz inequality

L^p norms of centered functions also satisfy Leibniz inequality

Connections to differential calculus and non-commutative geometry

Abstract

We show that certain symmetric seminorms on $\mathbb{R}^n$ satisfy the Leibniz inequality. As an application, we obtain that $L^p$ norms of centered bounded real functions, defined on probability spaces, have the same property. Even though this is well-known for the standard deviation it seems that the complete result has never been established. In addition, we shall connect the results with the differential calculus introduced by Cipriani and Sauvageot and Rieffel's non-commutative Riemann metric.