Consider an $n$-dimensional Euclidean Space. Consider orthants in that space. Each orthant occupies $\frac{1}{2^n}$ of the volume of an $n$-dimensional unit sphere. Let's call that a spherical orthant. Let's index spherical orthants by a binary $n$-tuple, e.g. for $n=4$, $(+,+,-,+)$ is the spherical orthant given by
$$x_1 > 0, \qquad x_2 > 0, \qquad x_3 < 0, \qquad x_4 > 0.$$
Now consider the "upper diagonal halfspace" given by
$$x_1 + x_2 + ... + x_n > 0.$$
What is the volume of the intersection of this "upper diagonal halfspace" with a spherical orthant, given as a fraction of the volume of that spherical orthant?
$\newcommand{\Vol}{\mathrm{Vol}}$
Let's see some examples. $D_n$ will be the upper diagonal half-space in $\mathbb{R}^n$. $\Vol(O_n)$ will be the volume of a spherical orthant.
$$(n= 2)$$
We have:
\begin{align*} \frac{\Vol((+,+)\cap D_2)}{\Vol(O_2)}=1 && \frac{\Vol((+,-)\cap D_2)}{\Vol(O_2)}=\frac12 \\ \frac{\Vol((-,+)\cap D_2)}{\Vol(O_2)}=\frac12 && \frac{\Vol((-,-)\cap D_2)}{\Vol(O_2)}=0. \end{align*}
$$(n=3)$$
We have:
\begin{align*} \frac{\Vol((+,+,+)\cap D_3)}{\Vol(O_3)}={}&1 & \frac{\Vol((+,+,-)\cap D_3)}{\Vol(O_3)}={}&y \simeq 0.78 \\ \frac{\Vol((+,-,+)\cap D_3)}{\Vol(O_3)}={}&y\simeq 0.78 & \frac{\Vol((-,+,+)\cap D_3)}{\Vol(O_3)}={}&y \simeq 0.78 \\ \frac{\Vol((+,-,-)\cap D_3)}{\Vol(O_3)}={}&1-y \simeq 0.22 & \frac{\Vol((-,+,-)\cap D_3)}{\Vol(O_3)}={}&1-y \simeq 0.22 \\ \frac{\Vol((-,-,+)\cap D_3)}{\Vol(O_3)}={}&1 - y \simeq 0.22 & \frac{\Vol((-,-,-)\cap D_3)}{\Vol(O_3)}={}&0, \end{align*}
with $y = 2 - \frac4\pi\arccos(\frac{1}{\sqrt3}) \simeq 0.78$.
$$(n= 4)$$
We have:
\begin{align*} \frac{\Vol((+,+,+,+)\cap D_4)}{\Vol(O_4)}={}&1 & \frac{\Vol((+,+,+,-)\cap D_4)}{\Vol(O_4)}={}&\frac{11}{12} \simeq 0.92\,\, \text{(3 pluses)} \\ \frac{\Vol((+,+,-,-)\cap D_4)}{\Vol(O_4)}={}&\frac12\,\, \text{(2 pluses)} & \frac{\Vol((+,-,-,-)\cap D_4)}{\Vol(O_4)}={}& \frac{1}{12} \simeq 0.08\,\, \text{(1 plus)}\\ \end{align*} $$\frac{\Vol((-,-,-,-)\cap D_4)}{\Vol(O_4)}=0.$$
Can you give the result for general $n$, and spherical orthants with $k$ times "+" ($0\leq k\leq n$) ?
Thank you,
Andreas