If $\gamma$ is a curve connecting $x$ and $y$, can we show $\int_0^1\left\|\gamma'(t)\right\|\:{\rm d}t\ge\left\|x-y\right\|$?

Question

Let $E$ be a $\mathbb R$-Banach space, $x,y\in E$ and $\gamma\in C^1([0,1],E)$. Can we show that $$\int_0^1\left\|\gamma'(t)\right\|_E\:{\rm d}t\ge\left\|x-y\right\|_E?\tag1$$ Clearly, by the mean value inequality, there is a $t_0\in(0,1)$ with $\left\|\gamma'(t_0)\right\|_E\ge\left\|x-y\right\|_E$, but that doesn't seem to be helpful.

Intuitively, interpreting the left-hand side of $(1)$ at the length of the curve $\gamma$, it is clear that there is no shorter curve then a straight line ...

Answer 1

We have

$$ \| y - x \|_E = \left\| \int_{0}^{1} \gamma'(t) \, \mathrm{d}t \right\|_E \leq \int_{0}^{1} \left\| \gamma'(t)\right\|_E \, \mathrm{d}t. $$