- #1
jostpuur
- 2,116
- 19
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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