The minimal displacement and extremal spaces

TL;DR

This paper demonstrates that certain classes of Banach spaces, including separable $C(K)$ spaces and preduals of $L_1$, are strictly extremal concerning the minimal displacement of Lipschitz maps, revealing new geometric properties.

Contribution

It establishes the strict extremality of these spaces for minimal displacement, extending understanding of Lipschitz map behavior in Banach space geometry.

Findings

Separable preduals of $L_1$ are strictly extremal.

Non-type I $C^*$-algebras are strictly extremal.

Every separable $C(K)$ space is strictly extremal.

Abstract

We show that both separable preduals of $L_{1}$ and non-type I $C^*$-algebras are strictly extremal with respect to the minimal displacement of $k$-Lipschitz mappings acting on the unit ball of a Banach space. In particular, every separable $C(K)$ space is strictly extremal.