  We consider composite-composite testing problems for the expectation in the
Gaussian sequence model where the null hypothesis corresponds to a convex
subset $\mathcal{C}$ of $\mathbb{R}^d$. We adopt a minimax point of view and
our primary objective is to describe the smallest Euclidean distance between
the null and alternative hypotheses such that there is a test with small total
error probability. In particular, we focus on the dependence of this distance
on the dimension $d$ and the sample size/variance parameter $n$ giving rise to
the minimax separation rate. In this paper we discuss lower and upper bounds on
this rate for different smooth and non- smooth choices for $\mathcal{C}$.
