Let $B$ be an affine manifold. A multi-valued function $K$ on $B$ is a collection of functions on an open cover $\{(U_{i},K_{i})\}$ such that on $U_{i}\cap U_{j}$, $K_{i}-K_{j}$ is affine linear. We say $K$ is convex if the Hessian $(\partial^{2}K_{i}/\partial y_{j}\partial y_{k})$ is positive definite for all $i$, in any, or equivalently all, affine coordinate systems $y_{1},\ldots,y_{n}$.

Given a pair $(B,K)$ of affine manifold and convex multi-valued function, the Legendre transform of $(B,K)$ is a pair $(\check{B},\check{K})$ where $\check{B}$ is an affine structure on the underlying manifold of $B$ with coordinates given locally by $\check{y}_{i}=\partial K/\partial y_{i}$, and $\check{K}$ is defined by