I want to perform an integration over those points on an $n$-sphere, the $n+1$ coordinates of which are nonnegative and ordered (say, from largest to smallest). (So, the points form a subregion in the first quadrant--of measure/area, obviously $\frac{1}{(n+1)!}$ that of the first quadrant itself.)
What ranges of integration on the $n$ parameters/angles should be employed? (I'm thinking in terms of $n=5$, but the question seems general in nature. Also, I'm thinking in the standard terms of radius 1, centered at the origin.)
In my 2002 paper https://arxiv.org/abs/quant-ph/0207181 (bot. p. 2), I claim that for $n=3$, the ranges to be employed are \begin{equation} \label{constr} {\pi \over 4} \leq \theta_{1} \leq {\pi \over 2}; \quad f(\theta_{1}) \leq \theta_{2} \leq {\pi \over 2}; \quad f(\theta_{2}) \leq \theta_{3} \leq {\pi \over 2}; \quad f(x) = {\cot^{-1} \Big( {\cos{{x}}}} \Big). \end{equation} This scheme obviously extends to $n \gt 3$.
However, no formal demonstration of this was presented, and, at this point in time, I do not readily see how to possibly justify this claim. (In a still prior paper https://arxiv.org/abs/quant-ph/9904101 p. 7, the same assertion was made, but in terms of half-angles--$\frac{\theta_1}{2},\ldots$)