Convexity in the interpolation spaces

TL;DR

This paper investigates how various convexity properties of norms behave under interpolation in complex and real interpolation spaces, showing that some properties interpolate while others do not, except in dual cases.

Contribution

It provides a detailed analysis of the interpolation behavior of different convexity properties in interpolation spaces, highlighting which properties are preserved and which are not.

Findings

Uniformly rotund and weakly uniformly rotund properties interpolate.

Locally uniformly rotund and weakly locally uniformly rotund properties generally do not interpolate.

Dual interpolation couples preserve the interpolation of certain convexity properties.

Abstract

In this work we study if the norms rotund, uniformly rotund, weakly uniformly rotund, locally uniformly rotund or weakly locally uniformly rotund interpolate in the complex or the real interpolation spaces. We will see that the properties uniformly rotund and weakly uniformly rotund interpolate, but the properties locally uniformly rotund or weakly locally uniformly rotund do not interpolate in general except for the dual interpolation couple.