*Ryotaro Tanaka (Niigata University) ryotarotanaka@m.sc.niigata-u.ac.jp

The James constant $J(X)$ of a Banach space $X$ measures the squareness of its
unit ball (or sphere). It is known, in general, that $\sqrt{2} \leq (X) \leq 2$ for any Banach
space $X$. An elementary (but important) geometric effect of James constant is that
uniformly nonsquare Banach spaces are characterized as those spaces with James
constant strictly less than $2$. In other words, the class of Banach spaces with James
constant $2$ just consists of Banach spaces that are not uniformly nonsquare. Then,
how is the class of James constant $\sqrt{2}$? This class contains all Hilbert spaces; for,
as an immediate consequence of the parallelogram law, $J(H) = \sqrt{2}$ for each Hilbert
space $H$. However, it does not coincide with the class of Hilbert spaces, since there
are non-Hilbert two-dimensional spaces $X$ with $J(X) = \sqrt{2}$.
The purpose of this talk is to present geometric characterizations of Banach spaces with James constant $\sqrt{2}$. From this, we see some geometric effects of James constant $\sqrt{2}$ on Banach spaces.