On smooth norms and analytic sets (Q1976605)

Let \(X\) be a Banach space which is not reflexive but has a separable dual. It is shown that \(X\) admits a smooth norm such that the set of norm-attaining functionals is a complete analytic set. In fact, if \(X\) is a non-reflexive space admitting a smooth norm (Fréchet sense), then there is an equivalent smooth norm such that for any Polish space \(N\) and any analytic set \({\mathcal A}\subseteq N\), \({\mathcal A}\) is the inverse image of the norm one, norm-attaining elements under a continuous map \(\varphi: N\to X^*\). A variant of Asplund's average norms is used.