I’ve been attempting to minimize this function to little avail.
$$A(x) = \left(\sum\limits_{i=1}^n|S_ix - t_i|^p\right)^\frac{1}{p}$$
Where $t$ and $S$ are lists of $n$ rational numbers and $p$ is an integer greater than 1.
I took the derivative of the function and set it equal to zero,
$$\frac{((\sum\limits_{i=1}^n|S_ix - t_i|^p)^{\frac{1}{p}-1}) (\sum\limits_{i=1}^n\frac{S_i|S_ix - t_i|}{S_ix - t_i}^p)}{n^\frac{1}{p}p}=0$$
Solving for $x$ is where I get stuck. We can divide both sides by $n^\frac{1}{p}p$ which eliminates that term, but I’m not sure how to progress further.
Perhaps there is a different method of approaching this problem that someone could help me understand.
Thanks for any future answers! :)