How to compute volume of a circle defined by L1 distance?

Question

In n dimension space, given a central $x=(x_1,x_2......x_n)$ and radius r, a circle C is defined as all point $y=(y_1,y_2,.....y_n)$ satisfy $ \sum_{i=1}^n\left\lvert y_i-x_i\right\rvert <= r$

I'd like to know if there exist any formula to calculate the volume of such circle?
For example, the Euclidean volume of circle defined by Euclidean distance can be calculated as shown http://en.wikipedia.org/wiki/N-sphere.

Answer 1

I don't think you should call it Euclidian volume, just volume, or L1 volume.

Calculate it with repeated integration. I.e. for each new dimension integrate from $r_n = 0$ to $r$ something like $V_{n-1}(r_n)*(r-r_n)$

Answer 2

I found the answer:

Volume of such ball is $$V_n^p(R)=\frac{(2R\Gamma(\frac{1}{p}+1))^n}{\Gamma(\frac{n}{p}+1)}$$ For my case, take p=1.