Convex open sets of $\Bbb R^m$: are they MORE than connected by polygonal paths parallel to the axis?

Question

I don't know if the following is true or not. I wish it were, since then I could use it to prove one other thing:

Let $U\subset \Bbb R^m$ be an open convex set and let $x,y\in U$. Denote $y=x+(a_1,\dots,a_m)$ (in the canonical basis). Then there is a permutation $\sigma:\{1,\dots,m\}\to \{1,\dots,m\}$ such that, if $\displaystyle x_k=x+\sum_{i=1}^ka_{\sigma(i)}e_{\sigma(i)}$, $k=1,\dots,m$ then the polygonal path $[x,x_1,x_2,\dots,x_m=y]$ is contained in $U$.

For example, if we consider an elipse (which is convex) and $x$ and $y$ as in the figure below then the path $P_1=[x,x+a_1e_1,x+a_1e_1+a_2e_2=y]$ doesn't work, but the path $P_2=[x,x+a_2e_2,y]$ does:

Here, $\sigma:\{1,2\}\to \{1,2\}$ is given by $\sigma(1)=2$, $\sigma(2)=1$.

Is the result true? I couldn't prove it neither to find a counterexample...

Answer 1

Counterexample in the picture where $U$ is the triangle.