This question is related to an orthant-type simplex in $\mathbb{R}^n$, which can be defined as
$$ S = \{x\in\mathbb{R}\_{+}^n: \sum\_{i=1}^nx\_i \leq 1\}=\overline{\mbox{conv}}(0,e\_1,\ldots,e\_n). $$
For a nonzero vertex $\bar x=e\_i$, we define the normal cone of $S$ at $\bar x$ as
$$ N\_{S}(\bar x)=\{y\in \mathbb{R}^n: y^T(x-\bar x)\leq 0\quad \forall x\in S \}. $$
My question is how to compute the 'volume' of this cone, for some reasonable notion of volume: this could be the area integral of the cone with the unit sphere, the Gaussian integral of the cone, or some other.


The second question is how to do this for a more general simplex
$$ S\_{\alpha} = \{x\in\mathbb{R}\_{+}^n: \sum\_{i=1}^n \alpha\_i x\_i \leq 1\}, $$
where $\alpha\_i>0$ for all $i$.


If there is no general formula in this case, it would be useful to know what type of invariants do these normal cones have, e.g. is the sum of its volumes (to some power) constant, regardless of $\alpha$?



Volume of normal cone of a simplex (at a vertex)
The volume of the cone is approximately $1/n$ for large $n$. Consider $C=N\_S(e\_n)$. 
In order for a vector $y$ to lie in $C$, it suffices that $y\cdot (-e\_n)\le 0$
and $y\cdot (e\_i-e\_n)\le 0$ for each $i<n$. 


That is, we require $y\_n\ge 0$ and $y\_n\ge y\_i$ for each $i<n$. 
In other words, we need $y\_n=\max(y\_1,\ldots,y\_{n-1},y\_n)$ and $y\_n\ge 0$. 


The first condition occupies a proportion $1/n$ of the space (by symmetry) and given that
the first condition holds, the second condition holds with probability close to 1. 


