Let $P$ is a polytope in $\mathbb{R}^n$ if $F$ is one of its faces of dimension $d$ then the dimension of its normal cone $\mathcal{N}(F)$ is $n-d$.\
This seems to be intuitively obvious but I can't write an explicit proof of it.
Such an statement is written as one of the properties of normal fans on Wikipedia (here) and as a reference it suggests the book "Lectures on Polytopes" by Ziegler, I searched a lot in his book but I couldn't find such an statement, maybe since I am really new in this field and I don't know much it is stated somehow indirectly and one can deduce this from some other statement.
I would be really grateful if someone can help me to find an explicit proof for this statement.