  Let $\mathcal{P}$ be a set of points in the plane, and $\mathcal{S}$ a
strictly convex set of points. In this note, we show that if $\mathcal{P}$
contains many translates of $\mathcal{S}$, then these translates must come from
a generalized arithmetic progression of low dimension. We also discuss an
application to the unit distance conjecture.
