Is there anyone familiar with this paper? It seems to me it contains "some" typos and even some small elementary mistakes, which makes my reading very slow. Of course the key reason for my slow reading is I wasn't familiar with Faltings and Chai's book, from which the paper use the notations and results.

Maybe I have some (or many) misunderstanding. I hope someone could help me clear up my confusions:

In Lemma 1.8, "the equation that cut out the cone at the vertex $(c,A(c))$ are $$x_0-A(c)\geq dA(\alpha(\sigma))(x)"$$ should the equations be $$"x_0-A(c)\geq dA(\alpha(\sigma))(x-c)" ????$$ "then the interior normal in the dual space $X^{\ast}_{\mathbb{R}}\oplus\mathbb{R}$ to the corresponding facet is $(1,-dA(\alpha(\sigma)))$", should this be "$(-dA(\alpha(\sigma)),1)$"???

In the same page, "Definition 1.10 We will denote the minimum of the functions $dA\alpha((\sigma))(x)$ in the above lemma by $\eta(x,c)$." Should we define $\eta(x,c)$ to be the maximum of the fuctions $dA\alpha((\sigma))(x)$????