Proposition 19.7.13F9F053A774FE

Proof. For compact sets $K,L,K',L'$, \[d_{\mathrm{Haus}}(K+L,K'+L')\leq d_{\mathrm{Haus}}(K,K')+d_{\mathrm{Haus}}(L,L').\] Indeed, given $x+y\in K+L$, select $x'\in K'$ and $y'\in L'$ at their respective distances from $x,y$; compactness ensures minimizers exist. The triangle inequality gives the stated bound on the distance to $K'+L'$. Interchange the pairs for the reverse bound. A Minkowski sum of two nonempty compact convex sets is nonempty, compact and convex. Hence this estimate proves continuity of the asserted map.