Let $E$ be a $\mathbb R$-Banach space, $v:E\to[1,\infty)$ be continuous and $v_i:[0,\infty)\to[1,\infty)$ be continuous and nondecreasing with $$v_1(\left\|x\right\|_E)\le v(x)\le v_2(\left\|x\right\|_E)\;\;\;\text{for all }x\in E,\tag1$$ $$v_1(a)\xrightarrow{a\to\infty}\infty\tag2$$ and $$av_2(a)\le C_1v_1^\theta(a)\;\;\;\text{for all }a>0\tag3$$ for some $C_1\ge0$ and $\theta\ge1$. Now, let $$\rho_r(x,y):=\inf_{\substack{\gamma\:\in\:C^1([0,\:1],\:E)\\ \gamma(0)\:=\:x\\ \gamma(1)\:=\:y}}\int_0^1v^r\left(\gamma(t)\right)\left\|\gamma'(t)\right\|_E\:{\rm d}t\;\;\;\text{for }x,y\in E$$ for $r\in(0,1]$.
Let $r\in(0,1]$, $k,\delta>0$ and $$B:=\left\{(x,y)\in E^2:\rho_1(x,y)<k\text{ and }\rho_r(x,y)\ge\delta\right\}.$$ How can we show that if $v_1^{r-1}(0)k<\delta$, then $B=\emptyset$?
I'm unsure how we should approach this task. However, let me note the following: Let $(x,y)\in B$ and $\varepsilon>0$. By definition of the infimum, there is a $\gamma\in C^1([0,1],E)$ with $\gamma(0)=x$, $\gamma(1)=y$ and $$\rho_1(x,y)\le\int_0^1v\left(\gamma(t)\right)\left\|\gamma'(t)\right\|_E\:{\rm d}t<\rho_1(x,y)+\varepsilon<k+\varepsilon\tag4.$$ In particular, since $v\ge1$, $$\left\|\gamma(t)-x\right\|_E\le\int_0^t\left\|\gamma'(s)\right\|_E\:{\rm d}s<k+\varepsilon\tag5$$ for all $t\in[0,1]$. Now, \begin{equation}\begin{split}\rho_r(x,y)&\le\int_0^1v^r(\gamma(s))\left\|\gamma'(s)\right\|_E\:{\rm d}s\\&\le\sup_{B_{k+\varepsilon}(x)}v^{r-1}\int_0^1v(\gamma(s))\left\|\gamma'(s)\right\|_E\:{\rm d}s\\&\le\sup_{B_{k+\varepsilon}(x)}v^{r-1}(k+\varepsilon)\\&=\left(\inf_{B_{k+\varepsilon}(x)}v\right)^{r-1}(k+\varepsilon),\end{split}\tag6\end{equation} but I'm not able to derive a contradiction from these observations. (Can we somehow extend $(6)$ to $\varepsilon=0$?)
EDIT: Another useful fact seems to be that, since $r-1\in(-1,0]$, we obtain from $(1)$ that $$v^{r-1}(z)\le v_1^{r-1}(\left\|z\right\|_E)\le v_1^{r-1}(0)\;\;\;\text{for all }z\in E\tag7.$$
EDIT 2: Please take note of my related question: Bound length of a curve in a ball.