I understand that for $\mathbb{R}$, a set $A\subseteq\mathbb{R}$ is convex iff for any two points $x,y\in A$, the point $kx+(1-k)y\in A$ for all $k\in[0,1]$.
Here’s my question. Does the same definition still hold for non-Archimedean ordered fields $\mathbb{F}$ extended from $\mathbb{R}$? I.e. a set $A\subseteq\mathbb{F}$ is convex iff for any two points $x,y\in A$, the point $kx+(1-k)y\in A$ for all $k\in\{z\in\mathbb{F}:0\le z\le 1\}$.