# On certain geometric properties in Banach spaces of vector-valued functions

## TL;DR

This paper establishes a general reduction theorem linking geometric properties of Banach spaces, such as octahedrality and lushness, to their stability under certain space constructions, leading to new and alternative results.

## Contribution

It introduces a unifying reduction theorem that connects stability of geometric properties under finite sums to stability under K"othe-Bochner space formation.

## Key findings

- Theorem applies to properties like octahedrality, lushness, and Daugavet property.
- Derived new results on octahedral and almost square spaces.
- Provided alternative proofs for known properties and their stability.

## Abstract

We consider a certain type of geometric properties of Banach spaces, which includes for instance octahedrality, almost squareness, lushness and the Daugavet property. For this type of properties, we obtain a general reduction theorem, which, roughly speaking, states the following: if the property in question is stable under certain finite absolute sums (for example finite $\ell^p$-sums), then it is also stable under the formation of corresponding K\"othe-Bochner spaces (for example $L^p$-Bochner spaces). From this general theorem, we obtain as corollaries a number of new results as well as some alternative proofs of already known results concerning octahedral and almost square spaces and their relatives, diameter-two-properties, lush spaces and other classes.