  We consider interaction energies $E_f[L]$ between a point $O\in
\mathbb{R}^d$, $d\geq 2$, and a lattice $L$ containing $O$, where the
interaction potential $f$ is assumed to be radially symmetric and decaying
sufficiently fast at infinity. We investigate the conservation of optimality
results for $E_f$ when integer sublattices $k L$ are removed (periodic arrays
of vacancies) or substituted (periodic arrays of substitutional defects). We
consider separately the non-shifted ($O\in k L$) and shifted ($O\not\in k L$)
cases and we derive several general conditions ensuring the (non-)optimality of
a universal optimizer among lattices for the new energy including defects.
Furthermore, in the case of inverse power laws and Lennard-Jones type
potentials, we give necessary and sufficient conditions on non-shifted periodic
vacancies or substitutional defects for the conservation of minimality results
at fixed density. Different examples of applications are presented, including
optimality results for the Kagome lattice and energy comparisons of certain
ionic-like structures.
