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James’ theorem fails for starlike bodies. (English) Zbl 0983.46016
A closed subset $$A$$ of a Banach space is said to be a starlike body provided $$A$$ has a non-empty interior $$\text{int }A$$ and there exists a point $$x_0\in \text{int }A$$ such that each ray emanating from $$x_0$$ meets the boundary of $$A$$ at most once. Since every convex body is a starlike body, one may ask whether the famous James’ theorem on characterization of reflexivity remains true when one replaces the word “convex” with the word “starlike” in this theorem. The authors disprove this conjecture by showing that in the Hilbert space $$\ell_2$$ there exist a $$C^\infty$$ smooth bounded starlike body $$A$$ and a one-codimensional subspace $$H\subseteq \ell_2$$ with the property that for no $$y\in\partial A$$ is the hyperplane $$y+H$$ tangent to $$A$$ at $$y$$.

##### MSC:
 46B20 Geometry and structure of normed linear spaces 46G20 Infinite-dimensional holomorphy
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