Find out the convex hull of the set $\left\{\pm \mathbf{u} \mathbf{u}^{T} \mid\|\mathbf{u}\|=1\right\}$ in a compact form ($u$ is a n-d vector)

Question

According to the answer from @Cloudscape

• The first step of finding the convex hull of a given set would be to visualize the convex hull and guess it.
• The second step would be to prove your guess contains the set of which you wanted to find the convex hull.
• Next, prove that your guess is convex.
• Finally, prove that any convex set containing the set will include your guess.

But I am struggling with the first step, I don't know how to visualize the given set.

Additionally, I also wonder the specific meaning of "compact form".

Answer 1

Visualization of set: A matrix $A$ belongs to that set iff

1. A is symmetric.
2. $a_{ii} > 0 $ & trace(A) = $1$
3. $a_{ij}^2 = a_{ii} a_{jj}$

Can you carry on from here?