an:01594753
Zbl 0983.46016
Azagra, D.; Deville, R.
James' theorem fails for starlike bodies
EN
J. Funct. Anal. 180, No. 2, 328-346 (2001).
00074072
2001
j
46B20 46G20
starlike body; convex body; James' theorem; characterization of reflexivity
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 \(y\).
L.Janos (Kent/Ohio)