Neighbors, everybody needs good neighbors

There's a rule which applies in LA (County, or maybe city, or maybe California) about how far your pool can be from the boundary to your neighbors. If you want a 6 foot deep pool, you have to be 6 feet away. But if you want to have an 8 foot deep pool you need to be 8 feet away. It's not hard.
However, if you want to, you can get closer than this by getting permission from the neighbors, in writing. Our neighbor is about 107, has 24 hour care, and possibly never gets out of bed. (We've met a couple of the carers, one of whom was in her 70s. So we wrote a nice letter (recorded delivery) to the neighbor (I had to use propertyshark.com to find out her name) asking if she would mind if we built an 8 foot deep pool 5 feet from the boundary. The cool thing is, if you don't get a reply that says no within 30 days, you can take that as a yes. So we sat back and prepared to wait.
A letter came back, 4 days later - through the post, which I though curious, as the nearest mailbox is considerably further from her house than our mailbox (I'm trying to use American words here, and getting confused. What's the equivalent of a postbox, where you post the mail for the postpeople to pick it up? And what's the thing that we have outside our house that hte postpeople leave post in? Are they the same word? How do know if I mean postbox or letterbox if it is the same word). It said "I do object to ...."
Oh well, back to the drawing board ....