What is the name of the geometric structure given by the equation:
$$x_1^n+x_2^n+x_3^n+\ldots +x_D^n=R$$
Or in short form, expressed as:
$$\sum_{i=1}^{D} x_i^n=R$$
where R is a constant.
And how to compute the volume of such an object??
For $n=1$, it is called a D-dimensional hyper plane and for $n=2$, it is called a D-dimensional hyper sphere; with special names for $D=0,1,2,3$.
I have to compute the volume of this hyper solid in D dimensions.
I have no clue as to how to progress. I can only solve till $n=2$. But not for $n$ in general.
Any ideas on how to progress??
For $n$ odd, the solution set of the equation will be unbounded and will have infinite volume.
For $n=p$ even, the set $\sum x_i^p = r $ equals the sphere of radius $r^{1/p}$ in the $L^p$ norm (the boundary of the ball of radius $r^{1/p}$ in the $L^p$ norm).
You can turn the level sets for arbitrary $p\geq 1$ into compact surfaces if you instead consider the equation $\sum |x_i|^p = r$.
The problem of finding the volume of these spheres has been discussed here: https://mathoverflow.net/questions/234314/surface-area-of-an-ell-p-unit-ball
It seems that computing a nice formula for the volume of these spheres might be impossible (in the same sense that computing a `nice' formula for the perimeter of an ellipse is impossible because the corresponding integral is elliptic).