Let $H$ be a Hilbert space. How can we describe the set $\{ x \in H \mid \|x-y\| = a \|x-z\| \},$ where $y, z \in H$ are fixed and $a > 0$? Geometrically how does it look like?
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Square the inner products and work things out. Then, for $a≠1$ and modulo computational mistakes: \begin{equation*} \parallel x-\frac{y-a^{2}z}{1-a^{2}}\parallel =c, \end{equation*} where $c$ is a constant.
Urgje
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Hint:
$\|x-y\|$ denotes that distance from the point $x$ to the point $y$.
$\|x-z\|$ denotes that distance from the point $x$ to the point $z$.
Paul
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Is there a connection with the Appolonius sphere? – Boo Han Sep 23 '14 at 08:48
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Sorry I don't know Appolonius sphere; – Paul Sep 23 '14 at 08:52
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However, you can view a Hilbert space as a metric space, especially $\Bbb R$. – Paul Sep 23 '14 at 08:53
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It look like the hyperplane parpendicular to the line joining $y$ and $z$ and dividing the line in the proportion $a:(1-a)$.
QED
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