Consider $S$ which is a two-dimensional surface with an induced metric from a normed space $(\mathbb{R}^3,\|\ \|)$. When $c$ is a curve of unit speed $c : [0,l]\rightarrow S$, i.e. $\| c'(t)\|=1$, s.t. $c(0)=p,\ c(l)=q$, then consider a variation $c : [-\delta,\delta ]\times [0,l]\rightarrow S$ with $c(s,0)=p,\ c(s,0)=q$. Then find a sufficient condition s.t. ${\rm length}\ c(0,\ )$ is minimizing in this variation.
Reference : Intrinsic geometry of surfaces in normed spaces - Burago and Ivanov
Define $$ T : D=[-\delta,\delta ]\times [0,l]\rightarrow S,\ T(s,a)=c_t(s,a)$$
When $\| T\|\geq 1$, then $T(D)$ is tangent to $\| \ \|$-unit sphere along $T(c_t(0,a))$ and $T(D)$ is outside.