A set $C$ is called convex, if for any two of its points $x$ and $y$ points of the form $tx+(1-t)y$ for $t\in[0,1]$ belong to $C$ as well. This can be easily interepreted as the condition that for any two points in the set, the line joining these points belongs entirely to the set.
On a convex set $C$ one can define a convex function $f:C\to\Bbb R$, that is a function such that $\forall x,y\in C, t\in[0,1]$ the condition $f(tx+(1-t)y)\le tf(x)+(1-t)f(y)$ holds. Is there an interepretion for this condition akin to one for the definition of convex sets? Best I can do is "image of a line is less than or equal (whatever that means) to the line connecting the images of the endpoints". Is there anything neater/cleaner?
This is equivalent to the epigraph of $f$, which is $\{(x,t):f(x)\leq t,x\in Dom(f)\}$, being a convex set.