Bounding distance to an intersection of semialgebraic sets

Question

As a continuation of my recent question " Bounding distance to an intersection of polyhedra ", I would like to pose a similar question about semialgebraic sets. Namely, given any two semialgebraic sets $P$ and $Q$, are there positive constants $C_{PQ}$ and $\theta_{PQ}$ such that for any point $x\in{\mathbb R}^n$ one has a bound $$\DeclareMathOperator{\dist}{dist} \dist(x,P\cap Q)^\theta\leq C_{PQ} \max(\dist(x,P),\dist(x,Q))? $$ Presumably such a bound would be related to the Lojasiewicz inequality.

Answer 1

You also need convexity for this to be true. Otherwise, a counterexample, e.g.: $$\DeclareMathOperator{\dist}{dist} \begin{split} P &= \{xy<1\},\\ Q &= \{(1-x)y<1\}, \end{split} (x,y)\in\Bbb R^2 $$ then for $x_n=(1/2, n)$ we have $$ \max \{\dist(x_n,P),\dist(x_n,Q)\} <1/2 $$ and yet $$ \dist (x_n, P\cap Q) \to \infty \text{ as } n\to\infty $$