I was looking at the definition of log-concavity:
A function $F:\mathbb{R}^n\rightarrow\mathbb{R}$ is said log-concave iff $F(x)\geq 0\forall x\in\mathbb{R}^n$ and $$F(x)^\lambda F(y)^{1-\lambda}\leq F(\lambda x+(1-\lambda)y)$$ $\forall > x,y\in\mathbb{R}^n$ and $\lambda\in[0,1]$.
And, according to Brascamp-Lieb,
If $F$ is a log-concave function in $\mathbb{R}^{m+n}$ and $F:(x,y)\rightarrow F(x,y)$ for $x\in\mathbb{R}^m$ and $y\in\mathbb{R}^n$, then $G(x)=\int\limits_{\mathbb{R}^n}F(x,y)dy$ is a log-concave function in $\mathbb{R}^m$. The convolution of two log-concave functions is log-concave.
This leads me to the following two questions:
Is there any definition of what would be the concavity of functions defined $\mathbb{Z}^d\rightarrow\mathbb{R}$?
Is there a discrete analogous of the two theorems?
When I say a discrete analogous I mean substitute $\mathbb{R}^{n+m}$ by $\mathbb{Z}^{n+m}$ and then, as a consequence state, that a discrete convolution of two log-concave functions is also log-concave.
