Radius of the largest enclosed ball in the convex hull of an algebraic variety 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".
 A: This is not a direct answer to your request for lower bounds; just some remarks.
Sometimes the center of the largest enclosed ball is called the 
Chebyshev center, and you can find literature under that name.
(But beware: the Chebyshev center sometimes means instead
the center of the smallest enclosing ball.)
Sometimes it is called the ball center.
Your $\mathcal{V}^c$ is a convex set. Finding the largest enclosed ball 
in a convex set is a convex optimization problem.
If you can approximate $\mathcal{V}^c$ with a convex polytope, then
finding the biggest ball is a linear programming problem.
E.g., these notes formulate that LP problem:
PDF download notes.
A better source is 

Stephen Boyd, Lieven Vandenberghe. Convex optimization. Cambridge. 2004.
  (PDF download book.)

They discuss the LP problem on p.148, and discuss
the problem for a general convex set p.417ff.
There they say,

Problem (8.16) is a convex optimization problem, since each function $g_i$
  is the
  pointwise maximum of a family of convex functions of $x$ and $R$, hence convex.
  However, evaluating $g_i$
  involves solving a convex maximization problem (either
  numerically or analytically), which may be very hard. In practice, we can find the
  Chebyshev center only in cases where the functions $g_i$ are easy to evaluate.

