Nevermind! I worked it out. This question seems really silly now.
I am working through the proof of the Kantorovich inequality on pages 6 and 7 of the following lecture notes: u.mich optimisation lecture notes There's one step of it that I'm really struggling to understand. Could someone please explain to me how one gets from the RHS of the last identity on page 6 to the first LHS inequality on page 7.
I also have a little difficulty understanding the diagram. My interpretation of the diagram on page 7 is that: For any convex-hull type combination of the points $a_1, a_2,\dots , a_n,$ with weights $\lambda_1, \dots, \lambda_n,$ we can find a $\gamma$ such that $\gamma \frac{1}{a_1} + (1-\gamma)\frac{1}{a_n} > \sum_{i=1}^n \lambda_i\left(\frac{1}{a_i}\right)$.
However, in the problem, aren't we simultaneously trying to ensure that for those same weights $\lambda_i$, that our $\gamma$ is also maximising $\gamma a_1 + (1-\gamma) a_n > \frac{1}{\sum_{i=1}^n \lambda_i\left(\frac{1}{a_i}\right)} $, since, ultimately we're trying to find an upper bound for $\beta$ from the equation on the bottom of page 6?? How does the diagram deal with this latter aspect?
Much thanks in advance!