I’m trying to gain some insight about a problem I’ve been thinking about recently. I have managed to bring it about to the following form:
Find $\min\limits_{x} \sum\limits_{i=1}^{n} |x-x_{i}|^p$, where $1\leq p\leq\infty$ (at $\infty$ we take the max norm), and $x_i$ are fixed scalars.
I already know this has a single unique solution due to strict convexity of the $p$-norm, but can I say something analytic about this solution?
I know that for $p=2$ this is the arithmetic mean of the number, for $p=1$ it is the median and for $p=\infty$ this is arithmetic mean of the max and min of the scalars $x_i$.
It should behave nicely as $p$ varies, but I can’t quite put my finger on it…
