Rewrite bounds of an optimization problem into a constraint

Question

I have a nonlinear convex optimization problem that I want to solve with an interior-point-method. To do this, I need to rewrite my bound $x \in \mathbb{R}^m$, with $x_1, \dots, x_m \leq 1$, into an equivalent differentiable constraint $c(x) \geq 0$, where $c : \mathbb{R}^m \to \mathbb{R}_0$. Does anybody have an idea how to rewrite it and would like to help me?