1. Suppose x,r that achieves the minimum has the property that x is not an eigenvector. Can you get a contradiction from this? In particular, can you perturb x to get a smaller r?

Now, the  hardest part of the problem is the last one, namely is to argue that the r that achieves the minimum is the magnitude of the largest eigenvalue in magnitude of A. Here is a trick which I think is useful here. First, show that the largest eigenvalue in magnitude of a positive matrix A with identical row sums s is actually s. Now given an (r,x) that achieve the optimum, can you use them to construct a matrix D such that D^{-1} A D is positive and has identical row sums?

4. You would like to show that two convex sets are identical. A typical strategy is to assume not, apply the separating hyperplane theorem, and obtain a contradiction.

5. The key here just to play around with the definition of a linear variety. You know that quantites like 2x – y or 3x-2y are contained it provided that x,y are contained in it. You can keep taking linear combinations of such things to get still different linear combinations that are contained in it.