I am reading Talagrand's seminal paper Concentration of Measure and Isoperimetric Inequalities in Product Spaces. Lemma 2.1.2 on Page 12 obtains the bound
$$
\int_\Omega \frac{1}{g} d\mu \int_\Omega g d\mu \leq a(t) = \frac{1}{2} + \frac{e^t + e^{-t}}{2},
$$
for any measurable function $g \colon \Omega \to [e^{-t}, 1]$, probability measure $\mu$ and $t \geq 0$.
To obtain this bound, the proof first observes that the functional $L(g) = \int_\Omega \frac{1}{g} d\mu$ is convex over the convex set $\mathcal{C}_b = \{g \colon e^{-t} \leq g \leq 1, \int_\Omega g d\mu = {\rm constant = b}\}$, and hence will obtain its maximum on a extreme point. Then the proof asserts the following, which I do not understand:
Assuming $\mu$ contains no atoms, it is well known that any extreme point of $\mathcal{C}_b$ takes only values $e^{-t}$ and $1$. Thereby, it suffices to show that for $0 \leq u \leq 1$, we have $(1 - u + u e^t)(1 - u + u e^{-t}) \leq a(t)$.
I would greatly appreciate any help or references on understanding why the above is true.
Edit: Following the comments, here is my intuition about the description of the extreme points from finite dimensions. If each $g$ was a finite-dimensional vector, then the constraint set $\mathcal{C}_b$ is an axis-parallel box intersected by a plane, and, every extreme point has $g_i \in \{e^{-t}, 1\}$ for all indices $i$. I don't know how to generalize this to the case where $g$ is a function.
Edit 2: Extending the above finite dimensional analogy, take $u$ to be the number of coordinates that are $e^{-t}$. Then the first sum is $\sum_i \frac{1}{g_i} = 1 - u + u e^{t}$ and the second sum is $\sum_i {g_i} = 1 - u + u e^{-t}$. I still don't know how to formally generalize this to $g$ lying in an infinite dimensional function space.