Is there a formal way to characterise strong convexity about the optimum value of a strictly convex function? I have an objective that looks something like this: $J(p,q) = \sum_{i=1}^{n}d(\pi_{i},q_{i}) + \alpha\sum_{i,j}w_{ij}d(p_{i},q_{j})$ where d is a jointly convex Bregman divergence (like KL or squared Euclidean) and $\pi_{i}$'s and $w_{ij}$'s and $\alpha$ are given parameter values. At the optimal value $(\bar{p}, \bar{q})$, neither the first term, nor the second term becomes zero (this comes from the practical usage of this objective). Is it possible to show strong convexity of this function around $(\bar{p}, \bar{q})$. My objective is to show positive definiteness of the Hessian around $(\bar{p}, \bar{q})$ and I think it is easier to prove strong convexity compared to proving positive definiteness of Hessian.