Let $\mathcal{V}\subset\mathbb{R}^n$ be a real compact algebraic variety. Let $\mathcal{V}^c$ be the convex hull of $\mathcal{V}$ and let us assume that $\mathcal{V}^c$ has nonzero n-dimensional Lesbegue measure.

Question: Are there any known lower bounds for the inner radius (=radius of largest enclosed ball) of $\mathcal{V}^c$ (for instance in terms of the polynomials that generate $\mathcal{V})$.

Informal remark: I am especially interested in cases when $\mathcal{V}$ is generated by only a few polynomials of "low degree".