Distance between convex set and non-convex set?

Question

So in http://en.m.wikipedia.org/wiki/Shapley%E2%80%93Folkman_lemma there is some talk about distance between a mintowksi sum and a convex set. But I couldn't get how distance is being defined. Can anyone help here?

Answer 1

In general, the distance between two sets $A$ and $B$ is defined as $$d(A,B)=\sup\limits_{a\in A}\inf\limits_{b\in B}\|a-b\|.$$ So for example if $A=\{0\}$ and $B=\{1,2\}$ then $$d(A,B)=\inf\{|0-1|,|0-2|\}=\inf\{1,2\}=1.$$

Answer 2

As far as I understand it, the maximal distance between a set $A \subset \mathbb{R}^d$ and its convex hull $H$ is defined as $\sup_{x \in A, y \in H} d(x,y)$, where $d(x,y)$ is a usual distance between points of $\mathbb{R}^d$, i.e. $\sqrt{(x_1-y_1)^2+\cdots+(x_d-y_d)^2}.$