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

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.

• Yes, you can find it by induction. Look at the case $n=2$, then $n=3$. – Quang Hoang Oct 27 '14 at 19:24
• Thank you. But I'm looking for a general solution for high dimension. I have some trouble in doing such induction. – xysheep Oct 27 '14 at 19:30
• Another hint for induction: It's $\frac{r^n}{n!}$. – Quang Hoang Oct 27 '14 at 19:33
• Doesn't the circle (under the L1 distance) look like a square? Just curious =) – étale-cohomology Jul 14 '17 at 2:53

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