I'm working on a problem my teacher asked me to check if I was interested, which is 'how to prove that Hausdorff Distance is strictly a distance function'. More specifically, how to prove that
$$D_H(A,B)\le D_H(A,C)+D_H(B,C)$$
where A,B,C are three finite measurable sets,
and $D_H(A,B)=max(d_h(A,B),d_h(B,A))$,where$$d_h(A,B)=max_{a\in A}(min_{b\in B} ||a-b||^2)$$$$d_h(B,A)=max_{b\in B}(min_{a\in A} ||a-b||^2)$$