Uniformly convex approximation of convex domain Suppose we have a bounded convex domain $\Omega \subset \mathbb{R}^n$. Can we approximate the domain from within by an increasing sequence of uniformly convex domains $\Omega_n$ converging to $\Omega$ in the Hausdorff distance? 
By a uniformly convex $X$, I mean that the principal curvatures of $\partial X$ have a positive lower bound (take it to be smooth). An equivalent definition is that for every $\epsilon >0$, there exists $\delta > 0$ such that for all $x, y \in X$ with $\|x-y\| \geq \epsilon$ the distance from $(x+y)/2$ to the boundary is at least $\delta$.
I want to apply an approximation argument to a proof about convex domains.  
 A: This is true. There is a smooth convex function $u:\Omega\to\mathbb{R}$ such that $u(x)\to\infty$ as $x\to\partial\Omega$; see below. The sum $v(x)=u(x)+\|x\|^2$ has the same properties, and is strongly convex. Since $v$ is also locally Lipschitz, it follows that the level sets 
$\Omega_t = \{x:v(x)<t\}$ are uniformly convex domains. The convergence $\Omega_t\to\Omega$ follows from the fact that   any compact set $K\subset \Omega$ is contained in $\Omega_t$ for $t>\sup_K v$.
The existence of $u$ as above requires a proof. It suffices to construct a convex $w:\Omega\to(-\infty,0)$ such that $w(x)\to0$ as $x\to\partial\Omega$; then $u=-1/w$ does the job since the function $\xi\to -1/\xi$ is convex and increasing on $(-\infty,0)$. Such $w$ is called a (smooth) convex exhaustion function for $\Omega$. Google search brought up Smooth exhaustion functions in convex domains by Zbigniew Blocki (free access), where the existence of $w$ is stated with a reference to Hörmander's book Notions of Convexity; so I suggest consulting that book as well.
