Divergence functions and their relation to metrical, equiaffine and symplectic structures on the manifold

Given a manifold $M$, a divergence function $D$ is a non-negative function on the product manifold $M \times M$ that achieves its global minimum of zero (with semi-positive definite Hessian) at those points that form its diagonal submanifold $M_x$. It is well-known that the statistical structure on $M$ (a Riemmanian metric with a pair of conjugate affine connections) can be constructed from the second and third derivatives of $D$ evaluated at $M_x$. In Zhang (2004) and subsequent work, a framework based on convex analysis is proposed to unify familiar families of divergence functions. The resulting geometry, which displays what is called ``reference-representation biduality'', completely captures the alpha-structure (i.e., statistical structure with a parametric family of conjugate connections) of the classical information geometry. This is the alpha-Hessian geometry with equi-affine structure. Here, we continue this investigation in two parallel fronts, namely, how $D$ on $M \times M$ a) is related to various Minkowski metrics on $M$; and b) generates a symplectic structure of $M$. On point a, a set of inequalities are developed that uniformly bounds $D$ by Minkowski distances on $M$. On point b, convex-induced divergence functions will be shown to generate a K\"ahler structure under which the statistical structure of $M$ can be modeled.