Geometric criteria for the sequential Dini property

Let $C$ be a closed pointed convex cone in a real Banach space $X$, and set $$q_C(x)=\max\{\operatorname{dist}(x,C),\operatorname{dist}(x,-C)\}.$$ We prove that every increasing weakly convergent sequence converges in norm if and only if every weakly null sequence $(x_n)$ satisfying $q_C(x_n)\to0$ is norm null. Equivalent conditions are a sandwich principle on relatively weakly compact sets and uniform equivalence of the $q_C$ and norm metrics on each weakly compact subset, with a modulus depending on the weakly compact set. These give intrinsic geometric criteria for the property in a characterization question posed by Jochen Glück in 2024. They distinguish the sequential property from normality, which requires a uniform lower bound for $q_C$ on the entire unit sphere. Rosenthal's theorem yields an equivalent criterion through $\ell^1$ basic sequences. In particular, on a space containing no copy of $\ell^1$, the property is equivalent to normality. A closed nonnormal cone with the property exists on a Banach space exactly when it contains $\ell^1$, with no generating requirement on the cone. The criteria also show failure of the property for the pointwise cones of $C^k[-1,1]$, $k\geq1$, and $W^{1,1}(-1,1)$.

GitHub repository: https://github.com/eulogica/sequential-dini-criteria