We know that for differentiable functions defined on $\mathbb{R}^k$ we can use Hessian matrix to determine its convexity. Moreover, the convexity of a discrete function defined on $\mathbb{N}$ can also be defined as $f(n+1) - f(n) \ge f(n) - f(n-1)$.
Now I was wondering is there any definition of convexity for a discrete function defined on $\mathbb{N}^k$?
One can define convex functions on an arbitrary set $E\subset \mathbb{R}^k$ as follows: $f:E\to\mathbb{R}$ is convex if for every $x_0\in E$ there exists a vector $v\in\mathbb{R}^k$ such that $$ f(x) - f(x_0) \ge v\cdot (x-x_0) \quad \forall x\in E $$ In words, this means that at each point of the domain, $f$ admits an affine minorant that agrees with $f$ at that point.
When specialized to $E=\mathbb{R}^k$ or $E=\mathbb{N}$, this definition agrees with those quoted above.