Distance has an important impact on many aspects of game-play. Distance is calculated based on the location of a character's capital. Diplomatic actions are restricted by distance, and is also an important modifier in several diplomatic actions. In war, it is important as it impacts the speed of armies and fleets, and it impacts many other aspects of game-play as well.
Diplomatic actions are limited by distance, a character's maximum diplomatic range is a distance of 700 from their nearest controlled province. Having the same religion, culture group and culture as the target of your action extends the diplomatic range by 100 each. Also some diplomatic actions will receive modifiers due to distance such as forming alliances. A character's ability to join a defensive pact is also limited by distance, a distance within 300 is required for same religion group and a distance within 400 is required for a different religion group. A character's level of threat is not impacted by distance, other characters will see the same level of treat no matter their proximity. Chances of a vassal joining a independence faction is also increased by distance, with the greater the amount of distance resulting in a higher chance of joining the independence faction. The Seduction focus from the Way of Life DLC allows characters to use the seduce diplomatic action, like all diplomatic actions it is limited by distance and distance also affects the frequency of seduction events a character will receive.
The most obvious impact of distance is the movement of armies and fleets. Armies moving to provinces farther away will take longer to travel to than moving to provinces that are a shorter distance away. There are two traits that impact the speed of armies. The first is the leadership trait organizer and the second is the Rulership focus trait administrator from the Way of Life DLC. Fleets are also affected by distance(How?) when they move through sea zones and ports.
Hiring mercenaries is also affected by distance as they have predefined capital provinces. If a character is not close enough to the mercenary's capital they will be unable to hire them, the distance penalty is reduced if the character shares a culture with the mercenary band.
The price of a merchant republic trade post is increased by 30% times the distance from the closet republic holding. A Silk Road trade post has the advantage that it can be built by non-merchant republic characters, however its upgrades are dependent on no other upgrade level being with in in a certain distance, with 100 distance for level 1, 200 distance for level 2, and 500 distance for level 3.
The titles that adventurers target are also dependent on distance. The casus belli that non-claimant adventurers use is required to target a title within in a distance of 800.