  Let $K\subset\mathbb{R}^d$ be a self-similar set generated by an iterated
function system $\{\varphi_i\}_{i=1}^m$ satisfying the strong separation
condition and let $f$ be a contracting similitude with $f(K)\subset K$. We show
that $f(K)$ is relative open in $K$ if all $\varphi_i$'s share a common
contraction ratio and orthogonal part. We also provide a counterexample when
the orthogonal parts are allowed to vary. This partially answers a question in
Elekes, Keleti and M{\'a}th{\'e} [Ergodic Theory Dynam. Systems 30 (2010)].
  As a byproduct of our argument, when $d=1$ and $K$ admits two homogeneous
generating iterated function systems satisfying the strong separation condition
but with contraction parts of opposite signs, we show that $K$ is symmetric.
This partially answers a question in Feng and Wang [Adv. Math. 222 (2009)].
