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If you look at this previous problem, I asked how to find the sum of all the areas between two taxicab geometry circles. However, upon learning about $\ell^p$ norms, I thought it would be pretty interesting to extend the problem to all $\ell^p$ norm circles, not just $\ell^1$ (taxicab).

If $p=1$, then the result has already been found (the total area is $\frac{5k^2-k-4}{2}$). If $p = \infty$, then each "circle" is just a square, and the area is also easily found (I'm too tired to think about it, but I think it would just be $4(k^2-1)$). Is there, however, a general formula for the area of each circle and the total area of the regions between circles in terms of $k$ and $p$; that is, what is the equation for the area of each overlapping region?

The area of an individual circle, if I did it correctly, is the area of a Lamé curve with $r = p$ and a radius of $k-n$ (see the linked problem), which equals $\displaystyle 4(k-n)^2\frac{(\Gamma(1+\frac{1}{p})^2)}{\Gamma(1+\frac{2}{p})}$. This can be reduced to $\displaystyle 2(k-n)^2 \frac{\Gamma(\frac{1}{p})^2}{p \Gamma(\frac{2}{p})}$ (see equations 41 and 42 here).

Here are some explanatory pictures:

$k=5, p=1$

k=5, p=1

$k=5, p=2$

k=5, p=2

$k=5, p=3$

k=5, p=3

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    added a simplification of the area... any ideas? I have not come up with any yet.2010-09-30
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    What is $n$ supposed to be?2010-09-30
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    FYI, in this context one usually uses lower-case l: $l^p$ or $\ell^p$.2010-09-30
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    @Nate: noted. will fix in a moment. @J.M. $n = 0, 1, \ldots , k-1$ (see the linked problem)2010-10-02
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    If I'm understanding what you want to do correctly, why can't you just subtract the total area of one Lamé curve from another?2010-10-02
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    @J. M. because they all have different centers.2010-10-02
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    I added pictures to make it more clear.2010-10-02

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The superellipses would seem to fit your bill, as long as $p < \infty$.

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    alright, so I would get $A=4k^2\frac{(\Gamma(1+\frac{1}{p})^2)}{\Gamma(1+\frac{2}{p})}$. As far as I know there is no nice way of displaying that. Is that the only way to find the area?2010-09-26
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    @Eugene: I don't really think you can get it any simpler than as a ratio of gamma functions. In any event, general expressions for areas/volumes of balls will always require the gamma function.2010-09-26
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    fair enough. then is there a way to find the total area correctly? the overlap is done in such a way that it isn't obvious how to figure it out...2010-09-27
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    That's already the expression for the total area; the trick is to just compute the area of a single quadrant (i.e. have the area integral range from 0 to $\pi/2$) and then multiply the result by 4.2010-09-27
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    I meant the total area of a bunch of them overlapping; see the question I linked to in the original post (I edited it)2010-09-28