Bounded sets and finite visibility (Q1183655)

Let \(S\) be a bounded subset of a Banach space which is uniformly smooth and uniformly rotund. It is shown that if \(S\) has the finite visibility property (i.e., given \(A\subset S\), \(\text{card} A<+\infty\), there exists \(x_ A\) in \(S\) such that for each \(x\) in \(A\), \([x_ A,x]\subset S\)) then the closure of \(S\) is star-shaped.