Abstract: The infimal Hausdorff dimension of all quasysimmetric (qs) images of a metric space X is called Conformal dimension of X. A subset X of a line is called qs thick if every qs self map of the line maps X to a set of positive measure. Bishop and Tyson asked if there are sets which are not qs thick but still have conformal dimension 1. We will answer this affirmatively by proving that middle interval Cantor sets of Hausdorff dimension 1 have also Conformal dimension 1 provided they are uniformly perfect. In general the main obstruction for minimizing the dimension of a space X is the existence of "sufficiently many" curves in X. If the time permits we will define for any curve family in a plane a dimension type quasiconformal invariant. We will discuss some examples and will illustrate some connections of our invariant with the Conformal dimension of X.