We sometimes need the distance between two countries in econometric models, for instance in gravity models. Bur for some countries, I think this measure could cause problem. Let’s take an example. If we want to model the volume of trade between country (i) and country (j), the economic theory says it will depend on the distance between (i) and (j). Let (i) be China, (j_1) Japan, and (j_2) India. The distance between (i) and (j_1) will be lower than between (i) and (j_2). Can we consider, though, that Japan is farther from China than India is?
So, rather than compute the distance between two head cities, it might be more accurate to compute the closest distance between the borders. If a border is shared by country (i) and country (j), then the distance should be zero.