Optimal Configuration for a Set of Points

Optimal Configuration for a Set of Points

Consider a set of $n$ points on the plane with positions
$\mathbf{p}_1,\dots,\mathbf{p}_n$, such that each point $i$ has at least
one neighbor $j$ at a distance of no more than $\lambda$ away from it
(i.e. $||\mathbf{p}_i - \mathbf{p}_j||\leq \lambda$).
The question is: how do you choose the positions of the points in order to
minimize their second moment, defined as:
$ U = \sum \limits_{i} ||\mathbf{p}_i - \bar{\mathbf{p}}||^2, $
where $\bar{\mathbf{p}}=\sum \limits_{i} \mathbf{p}_i$ is the barycenter
of the points.
Intuitively, I think that the points should be placed along a straight
line spaced $\lambda$ from each other, but I am not sure how to (dis)prove
this.
Thanks for any help.