# How to apply Hahn-Banach to the convex hull?

I am trying to understand the proof of Lemma 4.1.2 in Michel Talagrand's publication from 1995 on concentration inequalities (see below for the precise question statement):

A bit of context: Talagrand fixes a point $$x\in X$$ (he uses the notation $$X=\Omega$$) and a subset $$A\subset X$$, where $$X=X_1\times X_2\times\dots X_n$$ is the product space of arbitrary non-empty sets $$X_1,\dots, X_n$$. The $$\alpha_i$$ and $$t$$ are all supposed to be positive real numbers. He defines $$A_t^c$$ as follows:

My question. I understand why (4.1.4) implies (4.1.5). However, Talagrand says that "the converse follows from the Hahn-Banach theorem". How does it follow from the Hahn-Banach theorem?

Note: The problem can be slightly reformulated by saying that we want to prove that for $$t>0$$ and all $$M\subset\{0,1\}^n$$, we have that whenever $$\text{for all }\alpha\in]0,\infty[^n, \text{ there exists a } m\in M \text{ such that } \langle \alpha, m \rangle \le t\lVert \alpha\rVert_2,$$ then

$$\min_{m \in \text{Convex hull of } M} \lVert m \rVert_2 \le t.$$

In fact, if somebody can show this, then I will be able to prove the conjecture formulated by me yesterday.

To solve the problem you mention at the end you can argue in this way: $$\min_{m \in \mathrm{conv} (M)} \|m\|_{2}=\min_{m \in \mathrm{conv} (M)} \max_{\|\alpha\|_{2}\leq 1} \langle \alpha, m\rangle = \max_{\|\alpha\|_{2}\leq 1} \min_{m \in \mathrm{Conv}(M)}\langle \alpha, m\rangle \leq \max_{\|\alpha\|_{2}\leq 1, \alpha \in [0, \infty)^{n}} \min_{m \in M}\langle \alpha, m\rangle \leq t$$
The only nontrivial observation was used is min-max theorem, which says that if $$X, Y$$ are convex compact sets, $$f(x,y)$$ continuous, convex in $$x$$ and concave in $$y$$ then $$\min_{x \in X} \max_{y \in Y} f(x,y) = \max_{y\in Y} \min_{x\in X} f(x,y)$$. Choose $$f(x,y)=\langle x, y\rangle$$
• Amazing! Can't this be used to also directly prove my conjecture here as the last inequality seems like an equality to me if you consider the Bauer maximum principle ? (The right-hand side is just the minimum of norms over the convex hull of $M$.) – Maximilian Janisch Jun 26 at 15:30