Seven views on approximate convexity and the geometry of K-spaces

We study the interplay between the behaviour of approximately convex (and approximately affine) functions on the unit ball of a Banach space and the geometry of Banach K-spaces. Introduction This paper deals with the local stability of convexity, affinity and Jensen functional equation on infinite dimensional Banach spaces. Recall that a function f : D → R is said to be ε-convex if it satisfies f(tx+ (1− t)y) ≤ tf(x) + (1− t)f(y) + ε for all x, y ∈ D, t ∈ [0, 1]. If no specific ε is required we speak of an approximately convex function. Of course, any arbitrary function which is uniformly close to a true convex function is approximately convex. These will be called trivial or approximable. It may happen that there are no more: Hyers and Ulam proved in [18] that if D is a convex set in R, then for every ε-convex function f : D → R there exists a convex a : D → R such that sup x∈D |f(x)− a(x)| def = dD(f, a) ≤ C · ε, where C = Cn is a constant depending only on n. It is apparent that the papers [18, 16, 8, 9, 7, 12, 11, 13, 14, 3, 28] contain the complete story of Cn. As far as we know, the first connections between approximately convex functions and the geometry of infinite dimensional Banach spaces appear in [7] and [3]. In [7] it was proved that Lipschitz ε-convex functions are approximable on bounded sets of B-convex spaces, with the distance to the approximating convex function depending only on ε. (Recall that B-convexity means ‘having non-trivial type p > 1’.) That paper contains some counterexamples based on the fact that l1 is the Banach envelope of the spaces lp for all 0 < p < 1. In [3] it was remarked that every Banach space which is not a K-space (see Section 1 for precise definitions) admits a ‘bad’ (that is, non-approximable) ε-convex function defined on its unit ball. To be a K-space is a homological property of Banach 2000 Mathematics Subject Classification: 46B20, 52A05, 42A65, 26B25. The research of the first and second named authors is supported in part by DGICYT—project-BMF2001-0813. 1 2 FÉLIX CABELLO SÁNCHEZ, JESÚS M. F. CASTILLO AND PIER L. PAPINI spaces which is closely related to the behaviour of quasi-linear maps. It suffices to recall here that the spaces lp and Lp are K-spaces if and only if p 6= 1. With this background, let us explain the contents and the organization of the paper. The first Section is preliminary: we use the fact that l1 is not a K-space to obtain explicit examples of ‘bad’ approximately convex functions on infinite dimensional simplexes (the examples in [3, 11] are not explicit.) This leads to the question of whether K-spaces admit ‘bad’ ε-convex functions on their unit balls. The (affirmative) answer comes in Section 2, where we exhibit a non-trivial approximately convex function on the infinite dimensional cube (the unit ball of l∞). This solves the main problem raised in [3]. Having seen that the local stability of convexity does not hold in K-spaces, we prove in Section 4 that the local stability of affinity is equivalent to being a K-space: precisely, a Banach space X is a K-space if and only if for every ε-affine function f defined on its unit ball BX there exists a true affine a : BX → R such that dBX (f, a) ≤ A · ε, where A is a constant depending only on X. In Section 4 we prove a similar result for Jensen’s functional equation f (