How to express a set as an intersection of halfspaces

Question

I have a set S = {x $\epsilon$ $\mathbb R^n$| $x^Ty \le 1$, $\forall y \epsilon A$}

Now, I want to prove that this set is closed and convex. I know that expressing this set as an intersection of homogeneous halfspaces means that the set is convex. My problem is how do I start?

$\bigcap _{x}$ {X |$xy \le 1, \forall y \epsilon A$}

Would this be an intersection of halfspaces?

Answer 1

You already have expressed $S$ as an intersection of closed half-spaces.

It's $S = \bigcap_{y \in A} H_y$, where $H_y$ is the half-space defined by the inequality $x^T y \leq 1$ (where $x$ is the variable).

A slight technicality arises with $y = 0$, in which case $H_y$ isn't a half-space. But that's easy to deal with.