2
$\begingroup$

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.

$\endgroup$
4
  • $\begingroup$ Yes, you can find it by induction. Look at the case $n=2$, then $n=3$. $\endgroup$ – Quang Hoang Oct 27 '14 at 19:24
  • $\begingroup$ Thank you. But I'm looking for a general solution for high dimension. I have some trouble in doing such induction. $\endgroup$ – xysheep Oct 27 '14 at 19:30
  • $\begingroup$ Another hint for induction: It's $\frac{r^n}{n!}$. $\endgroup$ – Quang Hoang Oct 27 '14 at 19:33
  • $\begingroup$ Doesn't the circle (under the L1 distance) look like a square? Just curious =) $\endgroup$ – étale-cohomology Jul 14 '17 at 2:53
0
$\begingroup$

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 $V_{n-1}(r_n)*(r-r_n)$

$\endgroup$
0
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.