Further results on angular equivalence of norms

TL;DR

This paper explores properties preserved under angular equivalence of norms, showing that such norms share geometric features like uniform non-squareness and exposed points, and discusses dual norm relationships.

Contribution

It extends the understanding of angular equivalence by establishing the preservation of geometric properties and addressing an open problem on dual norms.

Findings

Angular equivalent norms share the property of uniform non-squareness.

Angular equivalence preserves exposed points of the unit ball.

Partial results on the dual norms of angularly equivalent norms are provided.

Abstract

Angular equivalence of norms is introduced by Kikianty and Sinnamon (2017) and is a stronger notion than the usual topological equivalence. Given two angularly equivalent norms, if one norm has a certain geometrical property, e.g. uniform convexity, then the other norm also possesses such a property. In this paper, we show further results in this direction, namely angular equivalent norms share the property of uniform non-squareness, and that angular equivalence preserves the exposed points of the unit ball. A discussion on the (equivalence of the) dual norms of angularly equivalent norms is also given, giving a partial answer to an open problem as stated in the paper by Kikianty and Sinnamon (2017).