If we assume that each client has a similar antenna, and is working on the same frequency, then we can simplify the math by simply stating that the signal strength, in decibels, falls off linearly with distance.
There are then various ways you might decide what the "best" compromise in placement is, but one is the centroid, which minimizes the sum of squared Euclidean distances to each client.
Of course given pathological cases it won't work so well. Consider if you have 10 clients in your home, and one on the moon, the centroid will be somewhere in cislunar space, and you'll end up with a WiFi AP that doesn't work for any of the clients.
In practice, real world conditions make this mathematical solution purely academic. Practical applications must consider:
- reflection off objects
- attenuation from walls
- interference from other APs
- noise from other electronic devices
- antenna directionality and polarization
These effects make the idealized free-space path loss significantly different from the real loss. And in practice, and especially on the crowded 2.4 GHz band, avoiding interference is at least as critical as getting a good signal.
In practice, AP placement is done by picking a location which seems pretty close to most of the clients, but which is also clear of walls and other obstructions, and which is physically accessible for the installation. Then measurements of signal quality are made in situ, and adjustments are made as necessary.
