Can the convex hull of the graph of a nondecreasing function be all of $\mathbb R^2?$ Is there a non-decreasing function $f:\mathbb{R}\to\mathbb{R}$ such that the convex hull of its graph covers $\mathbb{R}^2$?
Recall that $Graph(f):=\{(x,f(x))\mid x\in\mathbb{R}\}$ and $conv(A):=\{\sum_{i=1}^n t_i a_i\mid n\in\mathbb{N}, \forall i=\overline{1,n},\ t_i\ge0, a_i\in A,\ \sum_{i=1}^n t_i=1\}$ is the smallest convex set that contains $A$. 
 A: The convex hull of the graph of $f(x) = x^3$ covers $\mathbb R^2$.
For a point $(x,y)$ with $y > x^3$, take the line through $(x,y)$ and $(x-1, (x-1)^3)$. This is a line with positive slope that's above the graph of $f$ at $(x,y)$; however, $f$ grows faster than any linear function, so this line eventually hits the graph of $f$ again at $(x', x'^3)$ for some $x'>x$. 
Therefore $(x,y)$ lies on the line segment connecting $(x-1, (x-1)^3)$ and $(x', x'^3)$, so it's in the convex hull of the graph of $f$.
For a point $(x,y)$ with $y < x^3$, the same argument applies, by symmetry. (Then $-y > -x^3$, so $(-x,-y)$ is on a line segment with endpoints $(a,a^3)$ and $(b,b^3)$, and $(x,y)$ is therefore on a line segment with endpoints $(-a,-a^3)$ and $(-b,-b^3)$.)
A: I agree that the convex hull of $f(x)=x^3$ covers $\mathbb{R}^2$.
First note that in the first quadrant the set:
Above$=\{(x,y)\mid x>0,\ y\ge x^3\}$ is included in the convex hull because every line with positive slope through the origin intersects $x^3$
By symmetry -Above which is Below $x^3$ in the third quadrant is in the convex hull
The convex hull of Above$\cup$-Above covers the positive $y-$axis and below $x^3$ in the first quadrant. Therefore the first (and by symmetry) the third quadrants are in the convex hull and that is enough.
A: Define the rays $R_1 = \{(t,t):t\le 0\}, R_2 = \{(t,2t):t\ge 0\}, R_3 = \{(t,t/2):t\ge0\}.$ Note that the convex hull of $R_1\cup R_2 \cup R_3$ is all of $\mathbb R^2.$
Now we can choose sequences $P_n \in R_2, Q_n \in R_3,$ both sequences $\to \infty,$ such that the polygonal path
$$\tag 1 R_1 \cup[(0,0),P_1]\cup [P_1,Q_1]\cup [Q_1,P_2]\cup [P_2,Q_2] \cup [Q_2,P_3] \cup \cdots$$
is the graph $G_f$  an increasing continuous function $f$ on $(-\infty,\infty).$
Because $(0,0),P_1,P_2,\dots \in R_2$ and $(0,0),Q_1,Q_2,\dots \in R_3,$ both $R_2,R_3$ are subsets of the convex hull of $G_f.$ So is $R_3.$ Since the convext hull of $R_1\cup R_2 \cup R_3$ is all of $\mathbb R^2,$ the same is true for the convex hull of $G_f.$
