Show that $(A^o)^o = A$ if A is compact and convex.

Question

I am working on exercises for a convex analysis course and I am slightly stuck on the following;

Suppose that $A \subset \mathbb{R}^n$ is closed, bounded (compact) and conex. Define the set

$$ A^o = \left\{y \in \mathbb{R}^n : y^Tx \leq 1, \forall x \in A\right\} $$

Show that $A^o$ is convex and $(A^o)^o = A$.

Showing that $A^o$ is convex is easy; take $u,v \in A^o$ and $\theta \in [0,1]\subset \mathbb{R}$. Then for all $x \in A$ we have

$$ (\theta u +(1-\theta)v)^Tx = \theta u^Tx+(1-\theta)v^Tx \leq \theta + (1-\theta) = 1 $$

and so $\theta u +(1-\theta)v \in A^o$.

Also showing that $A \subset (A^o)^o$ is easy since for each $x \in A$ and $y \in A^o$ we have

$$ x^Ty = y^Tx \leq 1. $$

Now showing that $(A^o)^o \subset A$ is where I am getting stuck. The way I've been trying to show this is by contradiction.

I suppose that there exists $x \in (A^0)^0$ so that $x \notin A$. Then let $\xi \in A$ so that

$$ ||x-\xi|| = \text{dist}(x,A). $$

Such a $\xi$ exists since $A$ is compact. Now since $A$ is convex as well as compact we know that the hyperplane

$$ H = \left\{y \in \mathbb{R}^n : a^Ty =b \right\} $$

with $a = x - \xi$ and $b = \frac{a^Tx+a^T\xi}{2}$ is a strict separating hyperplane for the set $A$ and $\{x\}$ with

$$ x \in \left\{y \in \mathbb{R}^n : a^Ty >b \right\} $$

and

$$ A \subset \left\{y \in \mathbb{R}^n : a^Ty < b \right\}. $$

Now this is where I get fully stuck. I have tried every which way that I can think of to get a contradiction from here with no success.

How would you suggest going forward.

NOTE: Hints are also welcome, if I get to a proof with the help of a hint then I will post my final proof as the answer to this question.

Thank you in advance for any help.

Answer 1

edit: As per the comment of gerw, the claim is only true if $0\in A$.

It seems to me like you already have a good idea with using seperating hyperplanes.

The next step would be to think of $A^\circ$ not just of a set in $\mathbb R^n$ but as a set of hyperplanes/half-spaces.

You have shown that there are $a,b$ such that $$ A\subset \{y\in\mathbb R^n : a^\top y<b \}. $$ Since $0\in A$, this implies that $b>0$. We can change $a,b$ by scaling such that $b=1$. Then we obtain $$ a^\top y < 1 \qquad\forall y\in A $$ which implies $$ a \in A^\circ. $$ Since $a^\top x > 1$, this shows that $x\not\in (A^\circ)^\circ$. This is a contradiction to your assumption.