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Log-concavity of lattice-functions and convolution

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 ...