Say I have a collection of m non-linear inequalities, where each of the $i = 1... m$ inequalities has the form:
$g_i(x) \leq 0$ where $g_i: \mathbb{R}^n \rightarrow \mathbb{R}$ and $g_i(x)$ is nonlinear with respect to $x \in \mathbb{R}^n$
Let $D$ be the set of feasible points $x \in \mathbb{R}^n$ - that is:
$D = \{x \in \mathbb{R}^n | g_i(x) \leq 0 \forall i \}$
Is $D$ a polyhedron? And does it always have vertices? And lastly, does $D$ have any special properties if $g_i(x)$ is convex for each $i = 1...m$?