Consider $\mathbb R^2$ with the standard norm. Let $c,c'$ be points on the same circle centered at $0$ i.e $d(0,c) = d(0,c')$. Let $p$ be another point such that $d(p,c')<d(p,c)$. Consider $\gamma, \gamma':[0,1] \rightarrow \mathbb R^2$ the two rays from $0$ to $c$ and $c'$ respectively i.e. $\gamma(t) = tc$, $\gamma'(t) = tc'$.
I am trying to prove that $d(p, \gamma'(t)) < d(p, \gamma(t))$ for any $t \in (0,1]$. Intuitively this seems to be the case from all drawings I can draw, for example:
But I'm unable to find a rigorous argument.