Let's say I have $n$ assets and their returns are stored in a matrix $X \in \mathbb{R}^{m \times n}$ (i.e. I have $m$ returns for each of them.
The covariance matrix of the returns is $\Sigma \in \mathbb{R}^{n \times n}$.
I define a portfolio $w \in \mathbb{R}^{n}$ and I want that $\sum_{i=1}^n w_i=1$.
My goal is to find all the portfolios such that the volatility of the portfolio is some target $\sigma^*$.
So my problem looks like this:
Find all $w$ such that: $\sqrt{w' \Sigma w}=\sigma^*$.
I think that in most cases, I would have an infinity of solutions as long as $\sigma^*$ was chosen decently with regards to the assets available.
What algorithm could help me to find them all? How would the result be represented? I was thinking it should give me some kind of vector space.