Let $V$ be an $n$ dimensional space with sets of positive class vectors $P$ and negative class vectors $N$. The task is to find a vector $x$ such that AUC is maximized, based on ranking generated by computing distances between $x$ and $P,V$. So in a sense, $x$ is closer to $P$ than to $V$. It looks like this doesn't have a unique solution, but I'm curious if there is a really easy explicit solution to this, or a short algorithm? Surely this is a well known classical problem?