Let $C$ and $D$ be two non-empty, disjoint and convex subset of $\mathbb{R}^n$. Let $X=C-D$ be the difference set. Then, since $C\cap D=\emptyset$ so $0\notin X$.
To find the distance between $C$ and $D$, I came across a formulation as follows:
$$\mathrm{dist}(C,D)=\mathrm{dist}(0,X)=\underset{x\in > X}{\inf}\|x\|.$$
And my question is: how is the distance between $C$ and $D$ is equal to the distance between $0$ and $X$?
I suppose $C$ is same as $Y$ and $D$ is same as $Z$. Also what you call as the difference set $Y-Z$ is $\{y-z: y \in Y, z \in Z\}$, not the set theoretic difference. By definition $d(Y,Z)=\inf \{|y-z|: y \in Y, z \in Z\}=\inf \{|0- (y-z)|: y \in Y, z \in Z\}=\inf \{|0-u|: u \in Y-Z\}=d(0,X)$.