Distance between a point and the sublevel set of a convex function

Question

Let $f(x) : \mathbb{R}^n \rightarrow \mathbb{R}$ be a convex and differentiable function, and let $P$ be a point in $\mathbb{R}^n$.

Define a function $g(m): R \rightarrow R$ to be the distance between point $P$ and the sub-level set $ K_m = \{ x \in \mathbb{R}^n \mid f(x) \le m\}$, i.e., $g(m) = d(P, K_m)$. Is the function $g$ a continuous function? If not, is there any restriction on $f$ that will make $g$ continuous? Thanks!

Answer 1

HINT: the function $g$ is convex.

Indeed, we have

$$\lambda_1 K_{m_1} + \lambda_2 K_{m_2} \subset K_{\lambda_1 m_1 + \lambda_2 m_2}$$ since $f$ is convex, and

$$d(P, \lambda_1 K_{m_1} + \lambda_2 K_{m_2}) \le \lambda_1 d( P, K_{m_1}) + \lambda_2 d(P, K_{m_2} ) $$ since $Q\mapsto d(P, Q)$ is convex.

Answer 2

Another way to look at this is note that $ h(x, \lambda, m) = d(x, P) + \lambda (f(x) - m)$ is an affine function of $m$ and a convex function for $x$, for each given $\lambda$.

Hence $h_1(x, \lambda) = max_{\lambda \ge 0} h(x, \lambda, m)$ is a convex function of $m$ and $x$ since point-wise supremum preserves convexity, and $g(m) = \min_x h_1(x,m)$ is convex in $m$ since minimization preserves convexity.