Banach spaces with the weak diametral diameter two property

Abstract

We introduce and systematically study the weak diametral diameter two property (weak-DD2P), a new geometric property that lies strictly between the diametral diameter two property and both the diameter two property and the convex diametral local diameter two property. A necessary condition for the weak-DD2P is obtained through the structure of the extreme points of the dual unit ball, which leads to characterisations for several classical classes of spaces, including $C(K)$ spaces, $L_1$-preduals, unital uniform algebras, and (vector-valued) function algebras. We establish stability results under standard constructions such as absolute sums, K\"othe--Bochner spaces, and projective (symmetric) tensor products. Moreover, we provide complete descriptions of the weak-DD2P for vector-valued spaces of the form $L_1(\mu,X)$, $L_\infty(\mu,X)$, and $C(K,X)$. These results yield a wide range of new examples and show that the weak-DD2P exhibits a behaviour genuinely different from that of other diameter two properties.