Thank you for the answer! I'm glad that the result holds! I have a few questions though if you don't mind: 1. What does pos mean in the definition of the cone? 2. Can you explain why the cone is tiled by integer translations of $Z$? 3. How constructive is this process? If I gave you $p$ could you give an algorithm to determine $p'$ and the $p_i$'s?

@JonMarkPerry Is it trivially obvious that it holds or doesn't? Can you elaborate a little more please?

@PerAlexandersson I completely agree. If $P$ has IDP or is integrally closed then the decomposition is possible. Are there conditions when this happens? Is there a construction for the decomposition? What if it doesn't have IDP? I've looked here and they prove that $kP$ will have IDP for all $k \geq n - 1$, but I don't think that's enough to prove this. They also provide an example where $2P\cap\mathbb{Z}^n \neq P\cap\mathbb{Z}^n + P\cap\mathbb{Z}^n$, but for $P$ with $\dim(P) \geq 7$. Is there an easy counterexample in my case?