How to proof that a straight line can split a convex to at most two regions? I am self-studying the book "Concrete Mathematics". The authors state the statement: "A straight line can split a convex region into at most two new regions, which will also be convex"
1) How can one proof a statement like this?
2) What introductory book would you recommend to understand convex functions/geometry?  
 A: I presume you are talking about the plane ($\mathbb{R}^2$)?
The statement is a little loose in that there are really three pieces (to be
honest, these sorts of loose statements drive me nuts). Let $C$ be the set, then there are the parts of $C$ that are on the line, the parts of $C$ strictly on one side of the line and the parts of $C$ strictly on the other side of the line.
Suppose you have a line $L = \{x | \langle d, x \rangle = c \}$, where $d \in \mathbb{R}^2$ and $c$ is a number. 
If we let $L_> = \{x | \langle d, x \rangle > c \}$, $L_< = \{x | \langle d, x \rangle < c \}$, you can see that we can write the plane as the disjoint
union $\mathbb{R}^2 = L_< \cup L \cup L_>$, and so
$C = (C \cap L_<) \cup (C \cap L) \cup (C \cap L_>)$.
Since the intersection of convex sets is convex, each of these three parts is
convex.
If you prefer, you could split into two parts using $\le$ and $>$ instead.
I don't really have an introductory reference. My favourite is Rockafellar's "Convex Analysis". Another nice text is Boyd & Vandenberghe's "Convex Optimization". 
