Arc length In \(L_\infty\) Metric

Calculus Level 5

The \(L_\infty\) metric defined on the Euclidean plane is a metric where the distance between two points \((a, b)\) and \((x, y)\) is \(\max \{|x-a|, |y-b|\}\).

Consider the arc \((t, t^2)\) for \(t \in [-1, 1]\). What is the length of this arc in the \(L_\infty\) metric?


This problem is obviously inspired from the \(L_1\) metric.

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