Complex Bishop--Phelps for general convex sets

Lomonosov constructed a complex Banach space $X_1$ and a closed bounded convex set $S_1\subseteq X_1$ having no support points. Here a support point is a point $x\in S_1$ at which some nonzero complex-linear functional attains

Equivalently in his construction, the zero functional is the only functional whose modulus attains its supremum on $S_1$.

Consequently the real general-convex-set conclusion in Bishop phelps has no unrestricted complex analogue. There is no conflict with that local theorem: under its declared DC and relative Hahn--Banach assumptions it proves the complex result only for the closed unit ball, not for every closed bounded convex set.

Externally proved; not proved here. Lomonosov first takes the closed convex hull of the point evaluations inside a predual of $H^{\mathrm{\infty }}$. Lemmas 1--2 and Theorem 1 use powers, the maximum-modulus principle, a norm-preserving extension to $C(M)$ on the maximal ideal space, and Riesz representation to show that its support functionals form only the line spanned by the identity function. He then quotients the predual by the line spanned by evaluation at zero. The dual of the quotient is the annihilator of that evaluation, whose intersection with the preceding support-functional line is zero; Theorem 2 concludes that the quotient image $S_1$ has no support points.

This item records only that source boundary. It is not a dependency of any other item in this pair, and neither a citation nor the summary above is treated as a local proof of Lomonosov's analytic construction.