Let G be a graph with vertices $1,2,...,n$ and $F(x)=\displaystyle{\sum_{ij\in\rm{E(G)}}x_ix_j}$.
Let S be the subset of $\mathbb{R}^n$ given by $x_i\ge 0$, $\sum x_i=1$. We're interested in $\displaystyle{\max_{x\in S}}$ $F(x)$.
Why is any local maximum of F in the interior of S also a global maximum?
I was reading this paper, http://math.ca/cjm/v17/cjm1965v17.0533-0540.pdf. In the middle of the third page, there is a remark saying:
Any local maximum of F in the interior of S is also a global maximum.
I don't see why this is true. Is this obvious? Or well-known?