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Suppose that $x,y$ are some vectors in a Euclidean space and $a,b$ are some scalars. Is there any inequality to factor out $||x-y||$ from

$$||a x - b y || $$

like this:

$$ ||a x - b y || \leq ||x-y|| \cdot \text{something} $$

?

What if the metric is $d_{\infty}$? Namely if:

$$ ||x-y|| = \max_i |x^i - y^i | $$

where $x^i$ is just the $i-$th coordinate of $x$.

Rubi Shnol
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  • By $d_\infty$ you mean that you are working with bounded sequences and $|x-y|=\sup |x_n-y_n|$? (I don't think it will change anything, but this information seems to be missing in your post.) – Martin Sleziak Oct 25 '15 at 11:40
  • Valery: Maybe it would be reasonable adding such information to the post. – Martin Sleziak Oct 25 '15 at 11:44
  • This is kind of trivial, so I assume this is not what you're looking for: $|ax-by| = |a(x-y)+(a-b)y| \le |a| |x-y| + |a-b|\cdot|y| = \left(|a|+|a-b|\cdot\frac{|y|}{|x-y|}\right)|x-y|$. (On the other hand, if you have some estimate on $\frac{|y|}{|x-y|}$ in the situation where you want apply the inequality you're asking for, this would gives you at least something.) – Martin Sleziak Oct 25 '15 at 11:47
  • Valery: I have noticed that in the meantime. (I noticed that post among linked question and I have also noticed your comment there.) – Martin Sleziak Oct 25 '15 at 11:55

1 Answers1

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Since it's trivial to have $x-y=0$ and $|| ax-by||>0$ no such bound can be obtained.

Shahar Even-Dar Mandel
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