Consider the set of (smooth) function on a generic differential manifold (let's say without boundaries). What kinds of additional structures on the manifold are sufficient and/or necessary to be able to speak of the "convexity" of a function – besides making the manifold a convex set, ie giving it a flat affine connection?
The answer to this question and this paper discuss geodesic convexity, based on a connection or on a metric structure, but I was wondering if other alternatives have ever been discussed in the literature, especially non-metric- and non-connection-based ones.
Please give some references with your answers, so I can explore further!
These comments are necessarily speculative, since the question amounts to "Is there a non-metric structure giving rise to a notion of convxity?", and I'm arguing "probably not".
Think first of a real interval. Convexity is not detectable by the smooth structure alone: For example, if $I = (0, 1)$ and $f(x) = x$, then $g(x) = f(x^{2})$ is strictly convex, and $h(x) = f(\sqrt{x})$ is strictly concave. These examples all but imply that some notion of distance is essential for defining convexity.
Inspired by Didier's comment, on a path-connected space we might impose a structure, "For every pair of distinct points $p$ and $q$, there exists a unique disinguished path joining $p$ and $q$." (On a Riemannian manifold of non-positive curvature, such as a flat affine space, geodesics would be the standard choice.) For brevity, let's call this a DP structure.
Whatever the notion of distinguished, one expects an "extension principle" of the type: If $r$ lies on the distinguished path from $p$ to $q$, then the distinguished path from $p$ to $r$ is the restriction. (If this is not the case, then distinguished is not locally-defined.) That means that for each point $p$, there is a family of maximal distinguished paths through $p$, whose union fills the entire space, and such that two maximal distinguished paths through $p$ intersect precisely at $p$.
An automorphism of a DP structure would be a diffeomorphism inducing a bijection on the set of distinguished paths. On the assumption that a DP structure induces a notion of convexity on some class of real-valued functions, convexity of any particular function $f$ is invariant under automorphisms of the DP structure. But the examples above indicate how rigid an automorphism of our hypothetical DP structure must be: Effectively affine in some sense.
To emphasize, this isn't a definitive analysis: I'm not asserting that no "non-metrizable DP structure" exists. But offhand, I would not expect such a structure to exist.