Convex subsets of semialgebraic sets Let's say that a semialgebraic set  $S \subseteq\mathbb{R}^n$ is "thick" if there is a semialgebraic $S_1 \subseteq \mathbb{R}^n$ and a positive $\epsilon$ such that 
$$\bigcup_{p\in S_1}B(p,\epsilon) = S.$$
Here $B(p,\epsilon)$ is the ball in $\mathbb{R}^n$ of radius $\epsilon$ centered at $p$.
Question: Must a thick unbounded semialgebraic subset of $\mathbb{R}^n$ have convex open subsets of arbitrarily large volume?

This should be a counterexample: Let $S_1=\{(x,x^2) \in \mathbb{R}^2: \,\, x \in \mathbb{R} \}$ and $\epsilon >0$ sufficiently small.
