MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 $r(t)$?

share|cite|improve this question
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 $\sqrt n/3$.

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.