We define $H^{n}$ for the set of all compact subsets of $\mathbb{R}^n$. Define the metric $\Delta$ in $H^{n}$ as following.Let $A,B \in H^{n}$ then define
$d(x,B):= \min \lbrace d(x,y): y \in B \rbrace$
$d^{*}(A,B):=\max\lbrace d(x,B); x \in A \rbrace$
$\Delta(A,B):=\max \lbrace d^*(A,B),d^*(B,A) \rbrace$
We can check that $(H^n, \Delta)$ is a complete metric space.
My question is
- what is the distance $d^*(A,B)$ and $d^*(B,A)$ in which $A$ is the unit disk and $B$ is the unit square.
- What are $d^*(A,B)$ and $d^*(B,A)$ if $A$ is the unit circle and $B$ is the unit disk?
My calculations always lead to confusion since I think $A$ and $B$ are symmetric. Please feel freely helping me solve this problems.
Thank for reading!
Update: I edited my post since I remembered the wrong definition. This problem is in fact problem 6 page 259 in the book "Invitation to Dynamical System". In the solution guide(I have just found it), the author gave the result, but I do not understand it clearly. I still get confusion. Please help me. Sorry for the wrong definition. Thank you very much !