Small set in a large box

Let $K\subset\mathbb R^d$ be a compact convex set which is an intersection of half-spaces defined by at most two coordinates. Let $Q$ be the smallest axes-parallel box containing $K$. We show that as the dimension $d$ grows, the ratio $\operatorname{diam}Q/\operatorname{diam}K$ can be arbitrarily large. We also give examples of compact sets in Banach spaces, which are not contained in any compact contractive set.