- #1

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Let's denote the ordinary metric with [itex]d[/itex], so that [itex]d(a,b) = \sqrt{(a_1-b_1)^2 + (a_2-b_2)^2}[/itex], and then let [itex]e[/itex] denote some other metric.

Is it possible to define such [itex]e[/itex] that

[tex]

\sup_{x\in S_e(r)} d(x,0) = \infty

[/tex]

for all [itex]r>0[/itex], where

[tex]

S_e(r) := \{x\in\mathbb{R}^2\;|\; e(x,0)=r\}.

[/tex]

UPDATE: Mistake spotted. I didn't intend to define [itex]S_e(r)[/itex] so that it can be an empty set with some [itex]r[/itex]. Post #4 contains a new formulation for the original idea of the problem.

Is it possible to define such [itex]e[/itex] that

[tex]

\sup_{x\in S_e(r)} d(x,0) = \infty

[/tex]

for all [itex]r>0[/itex], where

[tex]

S_e(r) := \{x\in\mathbb{R}^2\;|\; e(x,0)=r\}.

[/tex]

UPDATE: Mistake spotted. I didn't intend to define [itex]S_e(r)[/itex] so that it can be an empty set with some [itex]r[/itex]. Post #4 contains a new formulation for the original idea of the problem.

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