Asymptotic geometry and Delta-points

Abstract

We study Daugavet- and $$\Delta$$ Δ -points in Banach spaces. A norm one element x is a Daugavet-point (respectively, a $$\Delta$$ Δ -point) if in every slice of the unit ball (respectively, in every slice of the unit ball containing x) you can find another element of distance as close to 2 from x as desired. In this paper, we look for criteria and properties ensuring that a norm one element is not a Daugavet- or $$\Delta$$ Δ -point. We show that asymptotically uniformly smooth spaces and reflexive asymptotically uniformly convex spaces do not contain $$\Delta$$ Δ -points. We also show that the same conclusion holds true for the James tree space as well as for its predual. Finally, we prove that there exists a superreflexive Banach space with a Daugavet- or $$\Delta$$ Δ -point provided there exists such a space satisfying a weaker condition.

Delta‐points and their implications for the geometry of Banach spaces

“…More precisely, the two following facts were obtained: (i)If a point $$x\in S_X$$ is a $$\Delta$$‐point, then $$\alpha \left (S(x, \delta) \right) = 2$$ for every $$\delta > 0$$ [7, Theorem 3.5] or [40, Corollary 2.2]. In particular, no asymptotically smooth point can be a $$\Delta$$‐point [7, Proposition 3.6]. That is, $$x$$ is not a $$\Delta$$‐point if $$\lim _{t \rightarrow 0} \bar{\rho }(t,x)/t = 0$$, where $$\bar{\rho }(t,x)$$ is the modulus of asymptotic smoothness at $$x$$ (see, e.g., [4, Definition 14.6.1]). (ii)If a point $$x^*\in S_{X^*}$$ is a $${\rm weak}^*$$ $$\Delta$$‐point, then every $${\rm weak}^*$$ slice …”

“…Let $$x^* \in S_{X^*}$$ be such that there exists $$x \in S_X$$ with $$x^*(x) = 1$$. Then $$x^*$$ is ($${\rm weak}^*$$) $$\alpha$$‐strongly exposed by [7, Corollary 3.4], hence not a $$\mathfrak {D}$$‐point by Lemma 5.2.$$\Box$$…”

“…In particular, $$\alpha (S(x, \delta)) = 2$$ for every $$\delta > 0$$ since $$y^*(x) = 1$$. Then, by [7, Corollary 3.4], $$x$$ is not an asymptotically smooth point (as the Kuratowski measure of the $${\rm weak}^*$$ slices of the dual unit ball defined by an asymptotically smooth point goes to 0 as $$\delta$$ goes to 0).$$\Box$$…”

“…(i) If a point 𝑥 ∈ 𝑆 𝑋 is a Δ-point, then 𝛼 (𝑆(𝑥, 𝛿)) = 2 for every 𝛿 > 0 [7, Theorem 3.5] or [ As every point in the unit sphere of a weak * asymptotically uniformly convex dual space is weak * quasi denting (see, e.g., the discussion following Corollary 4.7 in [7]), Theorem 5.1 immediately follows.…”

“…Let 𝑥 * ∈ 𝑆 𝑋 * be such that there exists 𝑥 ∈ 𝑆 𝑋 with 𝑥 * (𝑥) = 1. Then 𝑥 * is (weak * ) 𝛼-strongly exposed by[7, Corollary 3.4], hence not a 𝔇-point by Lemma 5.2.It is unclear whether either[7, Theorem 3.5] or[40, Corollary 2.2] admit analogs for 𝔇-points. Yet, we can provide stronger duality results for those points that will in particular give new information about asymptotically smooth points and asymptotically uniformly smooth spaces.…”