I am looking to find the ratio of all the areas of A1 (the region where $|x_i|> r$ $\forall x_i$ i={1...n}) with respect to the area/volume for the hypersphere with radius 1 in n-dimensions, $S_1$. Even if a simple solution isn't possible a relatively decent upper bound would be worthwhile.
In the 2D case depicted an upper bound is relatively simple by assuming A1 is part of the square with length $a = (1-r^2)^{0.5}$. Thus $A1 = \frac{a^2}{2}+V(\text{segment})<\frac{a^2}{2}$ and then the ratio of the area occupied by all A1s is $\frac{\sum A1}{V_2(S_1)} < \frac{4\frac{a^2}{2}}{\pi}$, where $V_n(S_1)$ is the volume of the n-dimensional unit sphere
I tried generalising this for n-dimensional spheres as
$a = (1- (n-1)r^2)^{0.5}$
$\frac{\sum A1}{V_n(S_1)} < \frac{2^n\frac{a^n}{n}}{V_n(S_1)}$
I am not sure this generalisation is correct/valid due the weird way high dimensional shapes begin to behave?
Also it breaks down for very small $r$ such that $1 <\frac{2^n\frac{a^n}{n}}{V_n(S_1)} $ which is logical given that my estimation of the A1 area is an overestimate, as such I was wondering if there is a more accurate way to estimate A1? I'm not sure if I need to find the area of conic segments (I am aware of the paper by Li regarding this volume).
I am also aware that this is only valid for $r<\sqrt{\frac{1}{d}}$
