  We study a multi-marginal optimal transportation problem with a cost function
of the form $c(x_{1}, \ldots,x_{m})=\sum_{k=1}^{m-1}|x_{k}-x_{k+1}|^{2} +
|x_{m}- F(x_{1})|^{2}$, where $F: \mathbb{R}^n \rightarrow \mathbb{R}^n$. When
$m=4$, $F$ is a positive multiple of the identity mapping, and the first and
last marginals are absolutely continuous with respect to Lebesgue measure, we
establish that any solution of the Kantorovich problem is induced by a map; the
solution is therefore unique. We go on to show that this result is sharp in a
certain sense. Precisely, we exhibit examples showing that Kantorovich
solutions may concentrate on a higher dimensional sets if any of the following
hold: 1) $F$ is any linear mapping other than a positive scalar multiple of the
identity, 2) the last marginal is not absolutely continuous with respect to
Lebesgue measure, or 3) the number of marginals $m \geq 5$, even when $F$ is
the identity mapping.
