Does the Kreĭn–Milman property imply the Radon–Nikodym property for arbitrary Banach spaces?

A Banach space has the Kreĭn–Milman property (KMP) if every nonempty closed, bounded, convex subset is the norm-closed convex hull of its extreme points, and the Radon–Nikodym property (RNP) if every countably additive vector measure of bounded variation that is absolutely continuous with respect to a scalar measure admits a Bochner integrable density (equivalently, every closed bounded convex set has slices of arbitrarily small diameter). Lindenstrauss proved in 1963 that the RNP implies the KMP. The converse — whether a Banach space in which every closed bounded convex set is the closed convex hull of its extreme points must have the RNP, with no order, duality, or decomposition hypothesis — has been open since the early 1970s and is sometimes called the Diestel conjecture. It is the general question underlying the 1981 Bourgain–Talagrand theorem that the two properties are equivalent for solid Banach lattices. The equivalence is also known for dual spaces, preduals of von Neumann algebras, strongly regular sets, spaces isomorphic to their square, and further special classes, but no proof or counterexample is known for the unrestricted statement, which remains open as of mid-2026.

Background

The classical theorem of Kreĭn and Milman states that every compact convex subset $C$ of a locally convex topological vector space is the closed convex hull of its extreme points; an extreme point of $C$ is a point $x\in C$ that cannot be written as $x=(y+z)/2$ with $y,z\in C$ and $y\neq z$. In an infinite-dimensional Banach space, closed bounded convex sets are generally not compact in the norm topology, so the classical theorem does not apply to them. One says that a Banach space $X$ has the Kreĭn–Milman property (KMP) if every nonempty closed, bounded, convex subset of $X$ is the norm-closed convex hull of its extreme points. The Radon–Nikodym property (RNP) originated in the theory of vector-valued integration. The Bochner integral is the standard extension of the Lebesgue integral to functions with values in a Banach space, built as limits of integrals of simple functions. A Banach space $X$ has the RNP if for every finite measure space and every countably additive set function $m$ of bounded variation, taking values in $X$ and absolutely continuous with respect to the underlying scalar measure, there is a Bochner integrable $X$-valued function $f$ with $m(A)=\int_A f\,d\mu$ for every measurable set $A$; this direct generalization of the classical Radon–Nikodym theorem can fail in infinite dimensions (for instance in $L^1[0,1]$, the integrable functions on the unit interval, or in $c_0$, the scalar sequences converging to zero under the supremum norm). The RNP has well-known equivalent geometric and analytic formulations; the most useful here is dentability. A slice of a subset $C$ of $X$ is a nonempty set of the form $\{x\in C: x^*(x)>\sup_C x^*-\alpha\}$ for a continuous linear functional $x^*\in X^*$ and $\alpha>0$, and a Banach space has the RNP exactly when every nonempty closed bounded convex subset of it has slices of arbitrarily small diameter; equivalently, every Lipschitz map from $[0,1]$ into $X$ is differentiable almost everywhere. Both the RNP and the KMP are invariant under equivalent renormings (replacing the norm by an equivalent one does not affect them), and the standard reference for the RNP and its many equivalences is the 1977 monograph of Diestel and Uhl. J. Lindenstrauss proved in 1963, through the theory of norm-attaining operators (linear maps whose supremum over the unit ball is achieved at some vector), that every Banach space with the RNP has the KMP. The converse direction became, and has remained, a central open problem of the geometry of Banach spaces: is the KMP equivalent to the RNP? According to Pavlakos and Petrakis (2011), this question — sometimes called the Diestel conjecture — has been open since 1973, both in the global form for spaces and in a local form for individual closed bounded convex sets (a set is an RN-set if every vector measure of bounded variation whose average range, the collection of normalized values $m(A)/\mu(A)$ over measurable sets $A$, lies in the set admits a Bochner density, and one asks whether being the closed convex hull of extreme points forces this). All known positive cases use extra structure. For dual (conjugate) spaces the equivalence was settled around 1975: Stegall established the foundational results on the RNP in conjugate Banach spaces, and Huff and Morris proved that a dual space with the KMP has the RNP. The source of the present question is the 1981 note of Bourgain and Talagrand, which proves that in a Banach lattice with solid norm — a Banach space carrying a compatible order under which it is a lattice and whose norm satisfies $\|x\|\le\|y\|$ whenever $|x|\le|y|$ — the RNP and the KMP are equivalent, by an argument based on order-dentability, a lattice-theoretic analogue of the slice-based dentability described above. Caselles (1988) gave a short proof and extended the equivalence to preduals of von Neumann algebras, the spaces of ultraweakly continuous functionals on weakly closed self-adjoint algebras of bounded operators. Schachermayer proved the equivalence for spaces isomorphic to their own square and, most influentially, showed in 1987 that it holds for strongly regular sets — a closed bounded convex set is strongly regular if it admits, for each $\varepsilon>0$, convex combinations of slices of diameter less than $\varepsilon$, and on such sets the KMP forces the RNP. Further positive classes include spaces embeddable in a space with an unconditional finite-dimensional decomposition, that is a sequence of finite-dimensional subspaces such that every vector of the larger space has a unique expansion as an unconditionally convergent series of components from them (James), some spaces with a shrinking basis, meaning a Schauder basis (a sequence of vectors whose finite linear combinations converge in norm to every vector of the space) whose coordinate functionals also form a basis of the dual space (López-Pérez–Mena), and the closed bounded convex subsets of the spaces $C(\omega^{\omega^k})$ of continuous functions on the ordinal powers $\omega^{\omega^k}$ of the first infinite ordinal $\omega$, equipped with the order topology (Pavlakos–Petrakis). Weaker neighboring properties, such as the convex point of continuity property (existence of relatively weakly open subsets of arbitrarily small diameter in every closed bounded convex set), fill out a web of implications whose exact relation to the KMP is only partially understood; the strongest general statement available remains that the KMP implies the RNP only under additional regularity such as strong regularity.

Problem Statement

Let $X$ be an arbitrary real (equivalently, complex) Banach space, with no lattice order, duality, decomposition, regularity, or renorming hypothesis. Suppose that every nonempty closed, bounded, convex subset of $X$ is the norm-closed convex hull of its extreme points (the Kreĭn–Milman property). Must $X$ have the Radon–Nikodym property — that is, must every nonempty closed, bounded, convex subset of $X$ have slices of arbitrarily small diameter? Since Lindenstrauss proved that the RNP implies the KMP for every Banach space, an affirmative answer is equivalent to the statement that the Kreĭn–Milman property and the Radon–Nikodym property are equivalent properties of Banach spaces. The question is settled either by a proof, valid for all Banach spaces, that the KMP implies the RNP, or by an explicit Banach space that has the KMP but fails the RNP. This is the general question underlying the Bourgain–Talagrand equivalence for solid Banach lattices; it must be answered without restricting the class of spaces (no dual, lattice, unconditional-decomposition, strong-regularity, or separability assumptions), and neither the statement nor its negation may be verified on a proper subclass.

References

1. Jean Bourgain and Michel Talagrand, 'Dans un espace de Banach reticulé solide, la propriété de Radon-Nikodým et celle de Kreĭn-Mil′man sont équivalentes', Proceedings of the American Mathematical Society 81 (1981), no. 1, 93–96. DOI: 10.1090/S0002-9939-1981-0589144-X, https://doi.org/10.1090/S0002-9939-1981-0589144-X
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5. R. E. Huff and P. D. Morris, 'Dual spaces with the Krein-Milman property have the Radon-Nikodým property', Proceedings of the American Mathematical Society 49 (1975), 104–108. DOI: 10.1090/S0002-9939-1975-0361775-9, https://doi.org/10.1090/S0002-9939-1975-0361775-9
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16. Ginés López-Pérez and Juan F. Mena, 'RNP and KMP are equivalent for some Banach spaces with shrinking basis', Studia Mathematica 118 (1996), no. 1, 11–17. Zbl: 0854.46016, https://www.impan.pl/en/publishing-house/journals-and-series/studia-mathematica/all/118/1/109411/rnp-and-kmp-are-equivalent-for-some-banach-spaces-with-shrinking-basis