Suppose we have an $n \times n$ real-valued matrix $A = (A_{ij})$. When is it the case that
- there exists a bivariate "convex" function $f: \mathbb{R}^2 \to \mathbb{R}$, and
- a permutation $\pi$ on $\{1,\dots,n\}$, and
- a collection of points $\{x_1,\dots,x_n\} \subset \mathbb{R}$, such that $ A_{\pi(i),\pi(j)} = f(x_i,x_j) $
Are things going to change considerably if we let $f: \mathbb{R}^d \times \mathbb{R}^d \to \mathbb{R}$ and $\{x_1,\dots,x_n\} \subset \mathbb{R}^d$?