  Let $V$ be a norm-closed subset of the unit sphere of a Hilbert space $H$
that is stable under multiplication by scalars of absolute value 1. A {\em
maximal vector} (for $V$) is a unit vector $\xi\in H$ whose distance to $V$ is
maximum $d(\xi,V)=\sup_{\|\eta\|=1}d(\eta,V)$, $d(\xi,V)$ denoting the distance
from $\xi$ to the set $V$. Maximal vectors generalize the {\em maximally
entangled} unit vectors of quantum theory.
  In general, under a mild regularity hypothesis on $V$, there is a {\em norm}
on $H$ whose restriction to the unit sphere achieves its minimum precisely on
$V$ and its maximum precisely on the set of maximal vectors. This
"entanglement-measuring norm" is unique. There is a corresponding
"entanglement-measuring norm" on the predual of $\mathcal B(H)$ that faithfully
detects entanglement of normal states.
  We apply these abstract results to the analysis of entanglement in
multipartite tensor products $H=H_1\otimes ...\otimes H_N$, and we calculate
both entanglement-measuring norms. In cases for which $\dim H_N$ is relatively
large with respect to the others, we describe the set of maximal vectors in
explicit terms and show that it does not depend on the number of factors of the
Hilbert space $H_1\otimes...\otimes H_{N-1}$.
