Let $S$ be a (basic) semialgebraic set, for which we know the constraints (i.e. we know the polynomial inequalities and equalities which define the set). Is it necessarily the case that the cone $C = \mathbb{R}^+ S$ is a semialgebraic set, and if so, can one determine the inequalities and equations which define $C$ (i.e. is there an algorithm to find the constraints)?

edit:

- what if $S$ is a spectrahedron?
- what if $S = \text{conv}(X)$ where $X$ is a real algebraic set?