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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 $\alpha$?

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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.

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  • $\begingroup$ Thank you. Although I don't understand the symmetry argument to obtain the $1/n$ volume. Do you use the fact that these cones partition $\mathbb{R}^n$ (and they are equivalent up to an orthogonal transformation)? If so, I am not sure about the probability close to 1 statement, but it is not so important, since I only care about nonnegative vectors in these normal cones. $\endgroup$
    – Cristóbal Guzmán
    Commented Jul 8, 2014 at 19:23
  • $\begingroup$ Right. The cones $C_j=\{y\colon y_j=max_{i=1}^n y_i\}$ have the same volume: the orthogonal transformation that simply rotates the string of coordinates by one place: $(y_1,\ldots,y_n)\mapsto (y_n,y_1,\ldots,y_{n-1})$ maps $C_j$ to $C_{j+1}$. $\endgroup$
    – Anthony Quas
    Commented Jul 8, 2014 at 23:10
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The first and second questions concern the volumes of either orthoschemes or general spherical simplices, and are quite hard in odd dimensions, and easier in even dimensions. The last question has a sort of an easy answer. In any case, wisdom may be gleaned from the wonderful book (Geometry II) of Alexeyevsky, Shvartzman, Solodovnikov and Vinberg in the Springer geometry series. The pages most directly relevant to your questions are here.

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  • $\begingroup$ Thank you, Igor. I'll look up the notes. Any keywords for the 'easy answer'? $\endgroup$
    – Cristóbal Guzmán
    Commented Jul 8, 2014 at 19:27
  • $\begingroup$ Just look at the formulas in the link, and you will see (an) answer... $\endgroup$
    – Igor Rivin
    Commented Jul 8, 2014 at 20:07

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