# How to prove that trangle inequality is satisfied in Hausdorff distance

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$$d_h(A,B)=max_{a\in A}(min_{b\in B} ||a-b||^2)$$$$d_h(B,A)=max_{b\in B}(min_{a\in A} ||a-b||^2)$$

• Hausdorff distance is a set distance which is computed, by the specifuc rule, from point (elementary) distance. But the latter could be various. That could be euclidean. But could be, for example, Manhattan. Or could be some nonmetric dissimilarity. Dec 5 '21 at 13:03
• Thanks for your helping. Maybe I should first try to solve the specific case of this problem, like Euclidean distance. I will change the question description. Anyway, thank you for reminding me of it. ^O^
– Hans
Dec 6 '21 at 1:50