The Eucledian norm of the gradient of a convex function

Question

Here is the problem I've been wondering about the past few days:

Let $g(x):\mathbb{R}^n \to \mathbb{R}$ be a convex function which is differentiable everywhere on its domain.

Is the set

$S := \{~ x ~ | ~ ||\nabla_x g(x)||_2 \leq c \}$

convex for some positive constant $c$?

Answer 1

Any smooth approximation of the $1$-norm will do: its gradient will have magnitude approximately $\sqrt{\|x\|_0}$, which is not a quasiconvex function. For example, take $$g(x,y) = \sqrt{x^2+1}+\sqrt{y^2+1}.$$ The norm of its gradient is $$\|\nabla g(x,y)\| = \left\|\begin{bmatrix}\cfrac x{\sqrt{x^2+1}} \\ \cfrac y{\sqrt{y^2+1}} \end{bmatrix}\right\| = \frac{x^2}{x^2+1} + \frac{y^2}{y^2+1},$$ and $\|\nabla g(x,y)\|\le 1$ if and only if $x^2y^2\le 1$.