The most distance preserving projection for a map?

[Naima]

What is the most distance preserving projection for a map?

[Falconius]

Generally the projections that are split, so like the one with boles for making a globe. Unfortunately the need to preserve distance often makes them less useful in practical terms, in addition there is always going to be some distortion.

[Hai-Etlik]

What is the most distance preserving projection for a map?

There's no projection that allows for measuring distances between arbitrary points that gives meaningful results outside of a restricted extent. For any projection, the results are just so bad that there's no practical point in comparing them.

Maps with more discontinuities have less distortion over their continuous sections, but the discontinuities themselves are just a different kind of distortion and measuring across them is non-obvious. If you can structure the map so that all the distances you want to measure are only over continuous portions of the map while bringing the distortion down to acceptable levels, then that might work for you, but tit's not necessarily possible to do so. Something like Dymaxion or Cahill-Waterman if you are only interested in land distances might work depending on how much distortion is acceptable, and then only if the land is arranged in a way that allows for such a projection.