For my homework I have to solve this.
Give a characterisation of the curves $\gamma : [0,1] \rightarrow \mathbb{R}^2$, with $||\cdot||_{1}$ so that $\gamma(0) = (0,0)$ and $\gamma(1) = (1,1)$. And the length of $\gamma$, denoted as $l(\gamma)$, is equal to 2.
What I've written:
$\gamma (t) = (g(t), h(t))$ with $g(t)=t$ and $h(t)= t$
$l(\gamma) = |1-0| + |1-0| = 2$
$l[0,a_{1}] = |a_{1} - 0| + |y_{1} - 0|$
$l[a_{1}, b_{1}] = |b_{1} - a_{1}| + |y_{2} - y_{1}|$
$l[b_{1},1] = |1 - b_{1}| + |1-y_{2}|$
And we know that : $ l[0,a_{1}] + l[a_{1}, b_{1}] + l[b_{1},1] > 2$
Is this enough? I'm not sure what else to write.
Hint
Let $C(\gamma)=\{\gamma : [0,1] \to \mathbb R^2 \mid \gamma(0) = (0,0), \gamma(1) = (1,1) \text{ and }l(\gamma) = 2\}$.
Prove that
$$C(\gamma) = \{\gamma : [0,1] \to \mathbb R^2 \mid \gamma(0) = (0,0), \gamma(1) = (1,1) \text{ and } \gamma_1, \gamma_2 \text{ are non decreasing}\}$$ where $\gamma_1, \gamma_2$ are the coordinates of the curve $\gamma$.