Let $P$ and $Q$ be polyhedra in ${\mathbb R}^m$ with a non-empty intersection. I believe there should exist a constant $C_{PQ}>0$ such that for any point $x\in {\mathbb R}^m$ the following inequality holds: $$d(x,P\cap Q)\leq C_{PQ}\cdot {\rm max} ( d(x,P),d(x,Q)).$$ Here $d(x,Y)$ denotes the Euclidean distance between $x$ and a subset $Y\subset {\mathbb R}^m$. Any suggestions about how to prove this?
(A related question: https://mathoverflow.net/questions/463696/)