I am looking for different examples (or even a complete characterization if this is possible) of continuous functions that are monotone along all lines. By that I mean functions $f\colon X\to\mathbb{R}$ such that for all $x,y\in X$ the function $g_{x,y}:[0,1]\to\mathbb{R}$, $\alpha\mapsto f(\alpha x + (1-\alpha)y)$ is monotone. $X\subset\mathbb{R}^n$ should be convex so that everything is well defined.
If $n=1$ and $X$ is an interval then obviously these are just the the usual monotone functions. Also for general $n$, every affine function is monoton allong all lines since $f(\alpha x + (1-\alpha)y)=\alpha f(x) + (1-\alpha) f(y)$. What other examples of functions are there satisfying this property?
Note that there are easy examples of functions $\big(\text{ like }f(x)=\Vert x\Vert^2\big)$ such that if you fix a certain $x$ (here $x=0$ ), then $g_{x,y}$ is monotone for all $y$. However $f$ is still not monotone allong all lines. Also this choice of $f$ shows that there seems to be no easy criterion, like looking at partial derivatives for example.