I have an odd map projection requirement, but I would like a world map projection where even very small countries, like Luxembourg or Belgium are still visible when also showing China and Canada on the same map. The idea is to use this map to show reach and engagement with the hue and therefore a map projection that minimises the area differences would be great as it maximises visibility for small countries for a given map size. Obviously angles, area and distances would be distorted, but as long as the country is recognisable (so not like some maps used in electoral mapping) that is fine. I guess rectangular maps would be great and the highest latitudes can be cut. Any ideas?
I am sorry, but this type of linear map can't exist. You said that you want to minimize differences, so you want to make small countries bigger. But unfortunately, big countries become bigger too (because of linear transformation), so that's not the solution.
I can only think of scaling individual portions of some standard projection. For example, you could scale Europe a little bit because of many small countries. You could also just scale the islands in the Oceania, but not the distances between them. Oops, it seems like we got too far into cheating, but as long as scaling countries to minimize differences is the only thing you want to do, there is no problem. Of course, your decision is the last.