  We give an algebraic definition of a Markowitz market and classify markets up
to isomorphism. Given this classification, the theory of portfolio optimization
in Markowitz markets without short selling constraints becomes trivial.
Conversely, this classification shows that, up to isomorphism, there is little
that can be said about a Markowitz market that is not already detected by the
theory of portfolio optimization. In particular, if one seeks to develop a
simplified low-dimensional model of a large financial market using
mean--variance analysis alone, the resulting model can be at most
two-dimensional.
