You also need convexity for this to be true. Otherwise, a counterexample, e.g.: P= {xy<1}, Q={(1-x)y<1}, x_n=(1/2, n), max dist(x_n,P),dist(x_n,Q) <1/2, dist (x_n, P\cap Q} \to \infty

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 $$ \dist (x_n, P\cap Q) \to \infty \text{ as } n\to\infty $$