I have this confusion related to the convexity of some functions. I was reading this paper. I have this graph consisting of nodes denoted by $x_i$. Some of these nodes are connected through edges. I am confused about the convexity of the following function
$F_1 = x^TLx$ where $x=[x_1,x_2,...x_N]$ N is the number of nodes and L is the Laplacian matrix
$F_2 = \sum_{i,j \epsilon E} |x_i-x_j|$ where E is the set of edges
$F_3 = \sum_{i,j \epsilon E} max\{|x_i|,|x_j|\}$
$F_4 = \sum_{i,j \epsilon E} ||x_i|-|x_j||$
It is said that except function $F_4$, all the other functions are convex. I want the intuition behind all this. Any help?
$F_1$: the graph Laplacian is positive semidefinite, hence the second derivative $F_1''$ is a positive semidefinite matrix, and $F_1$ is convex.
$F_2,F_3$: are sums of convex functions. The function $|\cdot|$ is convex (it is a norm), $\max(\cdot,\cdot)$ is a convex function, too.