Let $(X,\|\cdot\|)$ be a Banach space. Assume that $X$ is strictly convex and that its modulus of convexity $$ \delta_X(\varepsilon)=\inf\left\{1-\left\|\frac{x+y}{2}\right\| \colon \|x\|=\|y\|=1,\ \|x-y\|=\varepsilon\right\},\qquad 0\le\varepsilon\le 2, $$ is continuous on $[0,2]$.
Recall the following definitions.
- $X$ is strictly convex if for all distinct $x, y \in X$ with $\|x\| = \|y\| = 1$, we have $\|(x+y)/2\| < 1$.
- $X$ is locally uniformly convex (LUR) if for every $x \in X$ with $\|x\| = 1$ and every sequence $(x_n)$ in $X$ with $\|x_n\| = 1$, $$ \|x_n + x\| \to 2 \implies \|x_n - x\| \to 0. $$
- $X$ has the Kadets-Klee property (KK) if the weak and norm topologies coincide on the unit sphere $S_X = \{x \in X \colon \|x\| = 1\}$.
- $X$ has the Radon-Riesz property (RR; sequential Kadets-Klee): if a sequence $(x_n)$ in $X$ converges weakly to $x$ and $\|x_n\| \to \|x\|$, then $\|x_n - x\| \to 0$.
Note that $$ \mathrm{LUR}\;\Rightarrow\;\mathrm{KK}\;\Rightarrow\;\mathrm{RR}. $$
Questions. Are the following true?
- Must $X$ be locally uniformly convex (LUR)?
- Must $X$ have the Kadets-Klee property (KK)?
- Must $X$ have the Radon-Riesz property (RR)?
I am primarily interested in whether property (KK) holds under the given assumptions.
In this paper (Corollary 3), it is proved that property (RR) holds under these conditions. In that paper the Radon-Riesz property is obtained as a corollary of a more general theorem on weak convergence of measures. Can property (RR) be established directly, without using the theory of weakly convergent measures?
I am particularly interested in understanding whether the continuous modulus of convexity and strict convexity directly imply the stronger Kadets-Klee property, rather than just the sequential Radon-Riesz property. Any insights, references, or potential counterexamples would be greatly appreciated.
