Angular equivalence of normed spaces

TL;DR

Angular equivalence is a new, finer relation among norms on a fixed space that preserves key geometric properties and distinguishes between different finite-dimensional norms, unlike topological equivalence.

Contribution

The paper introduces angular equivalence as a new relation among norms, showing it is finer than topological equivalence and preserves geometric properties.

Findings

Angular equivalence is an equivalence relation among norms.

It preserves properties like uniform convexity and strict convexity.

No two ll^p norms on b^n are angularly equivalent.

Abstract

Angular equivalence is introduced and shown to be an equivalence relation among the norms on a fixed real vector space. It is a finer notion than the usual (topological) notion of norm equivalence. Angularly equivalent norms share certain geometric properties: A norm that is angularly equivalent to a uniformly convex norm is itself uniformly convex. The same is true for strict convexity. Extreme points of the unit balls of angularly equivalent norms occur on the same rays, and if one unit ball is a polyhedron so is the other. Among norms arising from inner products, two norms are angularly equivalent if and only if they are topological equivalent. But, unlike topological equivalence, angular equivalence is able to distinguish between different norms on a finite-dimensional space. In particular, no two $\ell^p$ norms on $\mathbb R^n$ are angularly equivalent.