Q:

Minkowski functional and strange theorem

I have a theorem that says the following: 
Let X be a normed space and $U\subset X$ a convex subset with $0 \in \text{int(U)}$, then we have: $U$ is absorbing and if $\{x;||x|| < \epsilon\} \subset U$, then we have $p_U(x)\le \frac{||x||}{\epsilon}$, where $p_U$ is the Minkowski functional.
The problem is that I have trouble to understand what the second part means. Is it refering to all $x \in X$. Or does it say for every x that is in this set is the relation with the Minkowski functional true? There are clearly quantors missing in this statement, so it is hard to understand what this means.
Does anybody here have an idea what this could mean?

A:

The theorem means that the inequality holds for all $x\in X$:
$$\bigl(\forall x\in X\bigr)\left(p_U(x) \leqslant \frac{1}{\varepsilon} \lVert x\rVert\right).$$
That inequality is easy to see since $r^{-1}\cdot \lVert\,\cdot\,\rVert$ is the Minkowski functional of $B_r(0)$ (very easy to see), and one has the implication
$$A \subset B \Rightarrow p_B \leqslant p_A,$$
which is also not difficult to prove.

