I am a physicist currently working on a question posed as part of an algebraic geometric description of a physical set: I did not find a question that is closely related to what I am searching for yet, but please feel free to just post the link below. My question concerns semialgebraic sets that are the convex hulls of affine varieties.

Imagine I am given an ideal $I_n$ of $k$ polynomials in $\mathbb{R}^n$ whose zero locus defines my variety. The variety is the image of all extreme points of a convex set in $\mathbb{R}^{m+n}$ given by the $n$-th elimination ideal of some ideal $I$.

As I experienced, it is not possible to recover the full set of Boolean combinations needed to describe the projection of my original set this way.(If you have any experience in that and know something that could help me, please let me know. I already tried Cylindrical Algebraic Decomposition but it turned out to be too complex.)

Weakening my expectations, is it correct that if I take a point $P\in\mathbb{R}^n$ that from my preliminaries cannot be part of the projection and demand the ideal $I_n$ to be prime, I can find a sign condition on each of the $k$ polynomials $f_i$ by evaluating $f_i(P)$? Then $Q\in\text{conv}(Z(I_n))$ if $\lnot f_i(Q)*_i0$ for all $i=1,...,k$ where $*_i$ is the inequality coming from $f_i(P)*_i0$.

I thought it is possible to use Separation Theorem from Convex Geometry. The only subtlety is how to deal with singular points?

I appreciate any kind of help!