The Daugavet property is equivalent to the polynomial Daugavet property

TL;DR

This paper proves that the Daugavet property is equivalent to the polynomial Daugavet property, resolving a long-standing open problem and extending geometric characterizations to polynomial topologies.

Contribution

It demonstrates the equivalence of the Daugavet and polynomial Daugavet properties and extends geometric characterizations to polynomial topologies, providing new examples of spaces with the property.

Findings

Daugavet property implies polynomial Daugavet property

Linear Daugavet centers are also polynomial Daugavet centers

Weak operator Daugavet property implies polynomial counterpart

Abstract

In this note, we prove that the Daugavet property implies the polynomial Daugavet property, solving a longstanding open problem in the field. Our approach is based on showing that a geometric characterization of the Daugavet property due to Shvidkoy, originally formulated in terms of the weak topology, remains valid when the weak topology is replaced by the weak polynomial topology. Using similar techniques, we further establish that every linear Daugavet center is also a polynomial Daugavet center, and that the weak operator Daugavet property implies its polynomial counterpart. As an application of the latter result, we present new examples of Banach spaces whose $N$-fold symmetric tensor products satisfy the Daugavet property.