In "Ming-Jun Lai, Larry L. Schumaker. Spline Functions on Triangulations. Cambridge University Press, 2007, p.72." we have:
A function $f$ defined on a triangle $T$ is said to be convex in the direction $u$ provided
$$\frac{f(w_3)-f(w_2)}{|w_3-w_2|} \ge \frac{f(w_2)-f(w_1)}{|w_2-w_1|}$$
for all ordered sets of points $w_1, w_2, w_3$ in $T$ lying on a line pointing in the direction of $u$. We say that $f$ is convex on $T$ provided it is convex in all directions. As is well known from calculus, if $f$ has two derivatives in the direction $u$, then this definition of convexity in the direction $u$ is equivalent to $D_u^2f(v)\ge 0$, all $v\in T$.
But All thing that we have about function convexity briefly is here: http://en.wikipedia.org/wiki/Convex_function.
I can't make a connection between definition of convex function and convexity of a function on a certain direction, also both of them with second directional derivative of function on a direction. How we can proof that usual definition of convex function and "We say that $f$ is convex on $T$ provided it is convex in all directions." are equivalence? What's proof of "If $f$ has two derivatives in the direction $u$, then this definition of convexity in the direction $u$ is equivalent to $D^2_uf(v)\ge 0$, all $v\in T$."?
A link to a comprehensive source is pleasured.