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Thread: Maximally separate points in concentric circles in the $L_1$ norm

  1. #1
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    Maximally separate points in concentric circles in the $L_1$ norm

    Given two concentric circles $X_1$ (green), $X_2$ (red) with radii $1$ and $2$, respectively, it is clear that the set of points that maximally separate the two circles in the $L_2$ norm is the circle with radius $\frac{3}{2}$ (blue).


    I would like to understand how to construct the set of points that maximally separate $X_1$ and $X_2$ in the $L_1$ norm. I understand the $L_1$ ball generally forms a diamond, but I am somewhat lost trying to figure out how to use this to maximally separate the two circles.

    Note: I define the maximum separator as the set of points that have two or more closest points, in the $L_p$ norm, on distinct circles ($X_1$ and $X_2$).

    Maximally separate points in concentric circles in the $L_1$ norm-screen-shot-2019-03-01-01.26.21.png
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  2. #2
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    Re: Maximally separate points in concentric circles in the $L_1$ norm

    Quote Originally Posted by skamoni View Post
    Given two concentric circles $X_1$ (green), $X_2$ (red) with radii $1$ and $2$, respectively, it is clear that the set of points that maximally separate the two circles in the $L_2$ norm is the circle with radius $\frac{3}{2}$ (blue).


    I would like to understand how to construct the set of points that maximally separate $X_1$ and $X_2$ in the $L_1$ norm. I understand the $L_1$ ball generally forms a diamond, but I am somewhat lost trying to figure out how to use this to maximally separate the two circles.

    Note: I define the maximum separator as the set of points that have two or more closest points, in the $L_p$ norm, on distinct circles ($X_1$ and $X_2$).

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    You need to give more clarity in your definition. What does it mean for a set to have two or more closest points on the distinct circles? I can't figure out what you are trying to get at.
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  3. #3
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    Re: Maximally separate points in concentric circles in the $L_1$ norm

    Quote Originally Posted by Walagaster View Post
    You need to give more clarity in your definition. What does it mean for a set to have two or more closest points on the distinct circles? I can't figure out what you are trying to get at.

    How do I define a separator?


    Let $\Delta$ be a set of points in the plane. I say $\Delta$ is a separator of $X_1$ and $X_2$ if any continuous function $f$ that passes through $X_1$ and $X_2$ must necessarily pass through $\Delta$.


    How do I define a maximum separator?


    Let $\Delta$ be any separator of $X_1$ and $X_2$. Pick a point $x \in \Delta$ and imagine growing a $\|.\|_p$-ball centred at $x$ (i.e. consider the ball $B_{\epsilon}(x):=\{z \mid \|z-x\|_p \leq \epsilon\}$). I call $\Delta$ a maximum separator, if $\forall x \in \Delta$ the following holds: $\forall \epsilon > 0$, $\exists m_1, m_2 \in B_{\epsilon}(x)$ and points $y_1, y_2 \in X_1 \bigcup X_2$, such that if $y_1$ is the point that minimizes $\|m_1 - y\|_p$ ( where $y \in X_1 \bigcup X_2$ ), then $y_1 \in X_1$, and equivalently if $y_2$ is the point that minimizes $\|m_2 - y\|_p$ ( where $y \in X_1 \bigcup X_2$ ), then $y_2 \in X_2$.


    An example of a separator that is not maximal?

    Lets take the extreme example of $X_1$ being a separator. Then clearly there exists $x \in X_1$ and some $\epsilon > 0$ such that for all points $z$ in $B_{\epsilon}(x)$ the points
    $y \in X_1 \bigcup X_2$ that minimize $\|z - y\|_p$ all lie on $X_1$ (i.e. $y \in X_1$ and $y \notin X_2$)
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