Could someone help me with this problem?
Let $C$, $D$ convex and closed sets such that the intersection is empty. I want to show that the function $f: \mathbb{R^n} \to \mathbb{R}$ defined by $f(x) = (1/2)(\|x-P_C(x)\|^2+\|x-P_D(x)\|^2)$, where $P_C(x)$ and $P_D(x)$ are the projections onto $C$ and $D$, respectively, has global minimum.
One way of thinking is that if I have a level set $L_r(f)$ noempty and compact for some real $r$ so I have the existence of the minimum. See that exists $L_r(f)$ noempty (just take any $r$ in the image) and closed (because $f$ is continuous) is not a problem. But I can't show that $L_r(f)$ is bounded.
Does someone have an idea to solve this problem?
If $C,D$ are not bounded, then it's possible there are points arbitrarily close to both sets. So in that case $f$ has no finite minimizer. For example, suppose $C = \{(x,y) \in \mathbb{R}^2 \ : xy \ge 1, x \ge 0, y \ge 0 \}$, and $D = \{(x,y) \in \mathbb{R}^2 \ : x \le 0\}$.
If either $C$ and $D$ are bounded, then indeed sublevel sets of $f$ must be bounded, and $f$ has a global minimum.