1. "pos" means "positive span"; that is, if $A$ is a subset of a real vector space, $pos(A)$ is the cone of linear combinations of elements of $A$ with nonnegative coefficients.

@JonMarkPerry: that the RHS is contained in the LHS is obvious, but it is not obvious that every point in the LHS decomposes as required in the RHS. Consider the following example. Let $P$ be the regular simplex inscribed in the unit cube, with vertices (0,0,0), (1,1,0), (1,0,1) and (0,1,1). The point $(1,1,1)$ is in $2P$ but it cannot be decomposed as the sum of two integer points in $P$. This is usually expressed as $P$ is not integrally closed or $P$ does not have the IDP property.

However, there is something wrong in your detailed definition of nice: "A triangulation of $P$ is nice if every $k$-dim simplex in the triangulation has a $k−1$-dimensional intersection with a $k$-dimensional face of P". This property, as written, fails for every $d$-simplex...

What fedja is describing is (a version of) the pulling triangulation, and indeed has the following "nice" property: "Every $k$-dim simplex in a pulling triangulation has (at least) a facet (that is, a $(k-1)$-dim face) contained in the boundary of $P$."