> "Since you can’t move without increasing the average distance, you must have started at the best spot."
> Technically this isn't a fully sound inference. All this proves is that you're at a local optimum. So you'd also need to know or show that there is only one optimum/the problem is convex/local=global or any variant.
You can't move away from the median without increasing the average distance over all doors, when you're standing on the median.
But that's just a special case of the also-easy-to-prove fact that you can't move away from the median without increasing the average distance over all doors, no matter where you're standing. You can show that by exactly the same argument used in the post - if you move away from the median, your total distance to all doors will increase, because you moved away from more than half of the doors.
This suffices to show that -- if there is a global optimum superior to the local optimum at the median door -- that global optimum cannot be located in the neighborhood of any point on the real number line, which is to say it cannot be located at any finite distance from the median. Or in other words, no such global optimum can exist.
> Technically this isn't a fully sound inference. All this proves is that you're at a local optimum. So you'd also need to know or show that there is only one optimum/the problem is convex/local=global or any variant.
You can't move away from the median without increasing the average distance over all doors, when you're standing on the median.
But that's just a special case of the also-easy-to-prove fact that you can't move away from the median without increasing the average distance over all doors, no matter where you're standing. You can show that by exactly the same argument used in the post - if you move away from the median, your total distance to all doors will increase, because you moved away from more than half of the doors.
This suffices to show that -- if there is a global optimum superior to the local optimum at the median door -- that global optimum cannot be located in the neighborhood of any point on the real number line, which is to say it cannot be located at any finite distance from the median. Or in other words, no such global optimum can exist.