I'm looking for examples of (classes of) problems with a non-convex, non-linear formulation, but convex feasible region.
That is, a problem of the sort: $$ \begin{array}{lll} \text{minimize} & c^Tx & \\ \text{subject to} & g_i(x) \le 0 & (i \in I) \end{array} $$ where at least some of the $g_i$ are non-convex.
At the same time, the feasible region $F = \{ x \in \mathbb{R}^n \ | \ g_i(x) \le 0, i \in I \}$ should be a convex set. Preferably $F$ is full-dimensional, in the sense of containing a small ball.
For $g_i$, both smooth and non-smooth examples would be interesting.
UPDATE: Bonus points for problems that have some practical relevance. I would like to use such an example as motivation for a method to solve these kinds of problems.