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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 !

  • 0
    Hint: there are points of the unit circle whose distance from the unit square is 1.2011-12-11
  • 3
    I guess you meant $\Delta(A,B):=\max \lbrace d^*(A,B),d^*(B,A) \rbrace$.2011-12-11
  • 0
    Thank, Christian Blatter. It's my mistake. It should be $\Delta(A,B):=\max \lbrace d^*(A,B),d^*(B,A) \rbrace$2011-12-11

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