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Let $X$ be a Banach space and $B_X$ be the unit ball.

Suppose that for each $\lbrace C_n\rbrace_{n=1}^\infty\subset B_X$ satisfying $C_n$ are closed convex and $C_n\supset C_{n+1}$ has a nonempty intersection.

Is it true that $X$ is reflexive?

3 Answers 3