An improvement of a theorem of Heinrich, Mankiewicz, Sims, and Yost

TL;DR

This paper enhances a theorem by replacing the concept of 'ideal' with 'almost isometric ideal', leading to new characterizations of geometric properties of Banach spaces.

Contribution

It introduces the concept of 'almost isometric ideal' in Banach spaces, refining previous results and enabling new characterizations of key geometric properties.

Findings

Replaces 'ideal' with 'almost isometric ideal' in the theorem.

Provides new characterizations of diameter 2 properties, Daugavet property, almost square, and octahedral spaces.

Abstract

Heinrich, Mankiewicz, Sims, and Yost proved that every separable subspace of a Banach space $Y$ is contained in a separable ideal in $Y$. We improve this result by replacing the term "ideal" with the term "almost isometric ideal". As a consequence of this we obtain, in terms of subspaces, characterizations of diameter 2 properties, the Daugavet property along with the properties of being an almost square space and an octahedral space.