The assumption that the densities are nonzero everywhere is supposed to rule out some kinds of solutions. The assumption of absolute continuity with respect to the Lebesgue measure is to make sure any event like “all three points forming an equilateral triangle” will be of null probability, while the nonzero density assumption is to make sure that the positions/arrival times aren’t compactly supported.
Without this assumption (and with the requirement of a maximum speed) there are alternative ways to solve the problem. For example, if the arrival times are compactly supported then in some solutions a “pivot player” around which everyone coordinates can determine some finite amount of time to wait before starting to move to ensure everyone else already used the radar and got his original position. There’s something similar with the original positions if we assume a speed limit on how fast people can move.
In short, this is an assumption that’s meant to make the problem harder, not easier. I mention it explicitly so people don’t give solutions which work in the special case of when some distributions are compactly supported.
The exact assumptions I mentioned in the problem are far from tight, and there are a variety of different solutions that work against adversaries of different capabilities. Any solution you end up finding will probably still work under more restrictive assumptions than the ones I outlined in the OP.
Responding to the points in your note:
The assumption that the densities are nonzero everywhere is supposed to rule out some kinds of solutions. The assumption of absolute continuity with respect to the Lebesgue measure is to make sure any event like “all three points forming an equilateral triangle” will be of null probability, while the nonzero density assumption is to make sure that the positions/arrival times aren’t compactly supported.
Without this assumption (and with the requirement of a maximum speed) there are alternative ways to solve the problem. For example, if the arrival times are compactly supported then in some solutions a “pivot player” around which everyone coordinates can determine some finite amount of time to wait before starting to move to ensure everyone else already used the radar and got his original position. There’s something similar with the original positions if we assume a speed limit on how fast people can move.
In short, this is an assumption that’s meant to make the problem harder, not easier. I mention it explicitly so people don’t give solutions which work in the special case of when some distributions are compactly supported.
The exact assumptions I mentioned in the problem are far from tight, and there are a variety of different solutions that work against adversaries of different capabilities. Any solution you end up finding will probably still work under more restrictive assumptions than the ones I outlined in the OP.