Assume that we take an $n$-simplex (with side length of 2 units) and place a unit $(n-1)$-sphere at each vertex. For $n=2$, half of a circle is enclosed inside the $2$-simplex. For $n=3$, solid angle of trapped volume for one $1$-sphere is $ 3 \arccos(\frac{1}{3}) - \pi $, so it is basically possible to calculate the total trapped volume (although it is still not quite clear to me).
Now, is there any closed-form equation to compute the total tapped volume of hyperspheres inside the n-simplex? If not, is there any computational approximation available for the trapped volume or solid angle in $n$ dimension?
- Sorry if the question is mathematically slippery, please feel free to edit the question. Unfortunately this is the result when a physicist thinks about math problems ;).
Certainly not an analytic answer or something that will work in very high dimensions, but if your dimension is around 5 or 10 or less then you can sample uniformly (in terms of flat cross section) from the simplicial cone at a vertex of the simplex, and then evaluate the Jacobian taking the flat cross section to the spherical cross section, and then your answer is (approximately, due to finite sample size) the average Jacobian multiplied by the volume of the flat cross section of the simplicial cone. Then you can divide by the surface volume of the sphere to get the proportion of the spheres covered.