Counterexamples in rotundity of norms in Banach spaces

TL;DR

This paper explores the distinctions between various forms of rotundity in Banach space norms, constructing specific examples to demonstrate the differences and answering open questions about norm properties.

Contribution

It provides the first constructions of norms that separate uniform rotundity, weak uniform rotundity, and uniform rotundity in every direction in Banach spaces, solving three open problems.

Findings

Distinguished between UR, WUR, and URED in Banach spaces.

Constructed dense norms with specific rotundity properties.

Created a smooth norm in c0 with a non-strictly convex dual norm.

Abstract

We study several classical concepts in the topic of strict convexity of norms in infinite dimensional Banach spaces. Specifically, and in descending order of strength, we deal with Uniform Rotundity (UR), Weak Uniform Rotundity (WUR) and Uniform Rotundity in Every Direction (URED). Our first three results show that we may distinguish between all of these three properties in every Banach space where such renormings are possible. Specifically, we show that in every infinite dimensional Banach space which admits a WUR (resp. URED) renorming, we can find a norm with the same condition and which moreover fails to be UR (resp. WUR). We prove that these norms can be constructed to be Locally Uniformly Rotund (LUR) in Banach spaces admitting such renormings. Additionally, we obtain that in every Banach space with a LUR norm we can find a LUR renorming which is not URED. These results solve three open problems posed by A.J. Guirao, V. Montesinos and V. Zizler. The norms we construct in this first part are dense. In the last part of this note, we solve a fourth question posed by the same three authors by constructing a $C^\infty$-smooth norm in $c_0$ whose dual norm is not strictly convex.