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Let $r(t), t\in [0,1]$ be a continuous piecewise $C^1$ curve on the plane where $r(0)=(0,0)$ and $r(1)=(1,0)$. The distance $|r(t)|$ is a non-decreasing function and the distance $|r(t)-(1,0)|$ is a non-increasing function. $r_m(t), t\in [0,1]$ defined below (a semi vesica piscis) is such a curve. $$ \begin{equation} r_m(t) := \begin{cases} \left(1-\cos\Big(\frac{2\pi}{3}t\Big),\sin\Big(\frac{2\pi}{3}t\Big)\right), & \forall t\in \Big[0,\frac{1}{2}\Big]; \\ \left(\cos\Big(\frac{2\pi}{3}(1-t)\Big),\sin\Big(\frac{2\pi}{3}(1-t)\Big)\right), & \forall t\in \Big[\frac{1}{2},1\Big]. \end{cases} \end{equation} $$

Question 1: Does curve $r_m(t)$ maximize the length of all admissible curves $r(t)$? Question 1 has been answered by Pietro Major below in the negative.

Question 2: Suppose $r(t)$ consists of $n$ straight line segments, what is the maximal length curve $r(t)$?

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up vote 3 down vote accepted

The length of these curves is unbounded. For a positive integer $n$ consider a triangular wave $f_n:[0,1]\to\mathbb{R}$ with support on $[1/3,2/3]$, making $n$ (isosceles) triangular impulses on $[1/3,2/3]$ with $\|f_n(x)\|_\infty=\frac{1}{6\sqrt n}$. The graph of $f_n$ is a curve satisfying the monotonicity constraint, with length larger than $\sqrt n/3$.

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Thank you @Pietro Major. That is clever. Can you take a look at the follow-up question, question 2? – Hans Sep 5 '13 at 13:19

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