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