Suppose you have an n-dimensional ball of radius R living within a d-dimensional space. Imagine the region in d-dimensional space consisting of all points that are a distance r outside the surface of that n-ball. What is the volume of this region for any n, d, R and r?
In other words, what is the volume of the Minkowski sum of the sets N and D, where N is the n-ball of radius R and D is the d-ball of radius r?
So take the special case of d=3 dimensions to begin with. For n=1 we have a cylinder of height 2R and cross-section with radius r plus two hemispheres on either end of a flat line (the 1-d sphere). The combined volume of the hemispheres is just $2*(1/2)*(4/3)*\pi$$r^{3}$. For n=2 the region desired is the cylinder with a cross-sectional area of radius R times the height 2*r plus the outer half of a donut. To find this semi-donut volume I understand we can use the second centroid theorem of Pappus-Guldin by using the centroid of the semicircle of radius r, which is $\frac{4r}{3\pi}$. Then we find the volume of the desired semi-donut region by taking the area of the semicircle of radius r and 'sweeping' it around the circumference of the circle of radius R + $\frac{4r}{3\pi}$. This gives the formula $2\pi(R+\frac{4r}{3\pi})*\frac{1}{2}\pi*r^{2}$.
For n=3 you could obviously find the desired volume by taking the difference in volume of the sphere of radius R+r and the sphere of radius R, but I want a method that can generalise to a hypersphere of any dimension n in a space of any dimension d. Can the approach using the Pappus-Guldin theorem be generalised? Alternatively, I'm thinking a solution using calculus must be possible, but I don't know how to find it. So is there a neat formula for this?
Thanks :)