Some convexity properties in direct integrals and K\"othe-Bochner spaces

TL;DR

This paper investigates convexity properties such as strict, local uniform, and uniform convexity in the context of direct integrals and K"othe-Bochner spaces, extending classical geometric analysis in these function spaces.

Contribution

It provides new insights into convexity properties in direct integrals and K"othe-Bochner spaces, broadening understanding of their geometric structure.

Findings

Characterization of strict convexity in direct integrals

Conditions for local uniform convexity in K"othe-Bochner spaces

Analysis of strong and very convex properties in these spaces

Abstract

The notion of direct integrals introduced by Haydon, Levy and Raynaud in 1991 is a generalisation of the well-known concept of K\"othe-Bochner spaces of vector-valued functions (using a family of target spaces instead of just one space). Here we will discuss some classical geometric properties like strict convexity, local uniform convexity and uniform convexity in direct integrals. We will also consider strongly convex and very convex K\"othe-Bochner spaces.