I have an optimization problem which requires that, for the vertices of the regular $n$-gon, $p_{1:n}$ stored as columns of a matrix $P$, $Px$ must be inside the bounded region of the $n$-gon.
For example, if $n=3$, then I have three vertices in $\mathbb{R}^2$ which form an equilateral triangle, and I want to know what set of points lie inside that triangle in terms of its vertices.
I am trying to write out these constraints more formally.
I considered writing that $\sum_{i=1}^n x_i = 1$, but I don't think these conditions work. At least, I wouldn't even know how to prove it.
How can I write the constraints in terms of $P$ and $x$ such that $Px$ is in the bounded region of the regular $n$-gon?
