This question is a sort of continuation of this other question of mine Tricky inequality on norms of vectors .
Let $v_0,v_1,w_0,w_1\in \mathbb{R}^2$ be four non-zero vectors such that:
- $v_0=(v_{0x},0)$, $v_1=(v_{1x},0)$ with $v_{0x}>0,v_{1x}>0$
- $w_0=(w_{0x},w_{0y})$, $w_1=(w_{1x},w_{1y})$ with $w_{0y}\ge 0,w_{1y}\ge 0$
- $\displaystyle{\frac{||v_1||}{||v_0||}> 1}$ and $\displaystyle{\frac{||v_1||}{||v_0||}> \max\{\frac{||w_1||}{||w_0||},\frac{||w_0||}{||w_1||}\}}$ and $\displaystyle{\frac{||v_1||}{||v_0||}> \max\{\frac{||v_1+w_1||}{||v_0+w_0||},\frac{||v_0+w_0||}{||v_1+w_1||}\}}$
Define $$v_t:=((1-t)v_{0x}+t*v_{1x},0),\quad w_t:=(1-t)w_0+t*w_1$$ for every $t\in [0,1]$.
Is it true that the following two inequalities
$$\frac{||v_{t_1}||}{||v_{t_0}||}\ge \max\{\frac{||w_{t_1}||}{||w_{t_0}||},\frac{||w_{t_0}||}{||w_{t_1}||}\} \quad \textit{and}\quad \frac{||v_{t_1}||}{||v_{t_0}||}\ge \max\{\frac{||v_{t_1}+w_{t_1}||}{||v_{t_0}+w_{t_0}||},\frac{||v_{t_0}+w_{t_0}||}{||w_{t_1}+w_{t_1}||}\}$$
are always satisfied for every $t_0,t_1\in [0,1]$, $t_0<t_1$?
I can't find any counterexample, but I either can't prove the claim
