The uniform strong diameter two property

We study a uniform version of the strong diameter two property. In particular, we find a characterisation that does not involve ultrafilters and we use it to provide some examples of spaces with this uniform property that do not follow from previously known results.

💡 Research Summary

The paper introduces and studies a uniform version of the strong diameter‑two property (SD2P) for Banach spaces, called the uniform strong diameter‑two property (USD2P). A Banach space X is said to have USD2P if, for every free ultrafilter U on ℕ, the ultrapower X_U possesses the SD2P. While it is immediate that USD2P implies SD2P (because X embeds as an almost‑isometric ideal in any ultrapower and SD2P is stable under such ideals), the converse fails, as shown in earlier work (Theorem 4.6 of