Problem Given N 3-dimensional points which are {$p_1,p_2,..,p_n$} where $p_i = (x_i,y_i,z_i) $ . I have to find the value of the formula
$$ \sum \limits_{i=0}^n \sum \limits_{j=i+1}^n \frac{ \mid \alpha*(x_i-x_j+\beta*(y_i-y_j)+\gamma*(z_i-z_j)+\delta) \mid }{ \sqrt{ (x_i-x_j)^4+(y_i-y_j)^4+(z_i-z_j)^4} } $$
for some given $ \alpha, \beta, \gamma, \delta $. all numbers are between 1 and M ( = 100).
I need an efficient method for the reduction of this formula to something which is easy to calculate.
So far, I have recognized that the above expression is the normal component of vector joining i and j from the the plane formed by $ \alpha, \beta, \gamma, \delta $ divided by the square of the distance between i and j in $l_4$ norm.
Please give any idea about how to reduce it to simpler standard form which can be calculated efficiently.
EDIT: By transforming and rotating the coordinate system such that the plane formed by $\alpha , \beta ,\gamma$ and $ \delta $ is represented by $z=0$, problem reduces to finding
$$ \sum \limits_{i=0}^n \sum \limits_{j=i+1}^n \frac{ \mid z_i-z_j \mid }{ \sqrt{ (x_i-x_j)^4+(y_i-y_j)^4+(z_i-z_j)^4} } $$