Locally minimal sets for conformal dimension

We show that for each 1 ≤ α < d and K < ∞ there is a subset X of R^d such that dim(f(X)) ≥ α = dim(X) for every K-quasiconformal map, but such that dim(g(X)) can be made as small as we wish for some quasiconformal g, i.e., the conformal dimension of X is zero. These sets are then used to construct new examples of minimal sets for conformal dimension and sets where the conformal dimension is not attained.