What is the farthest distance anyone on the earth can be away from another person before they begin getting closer again?

What I mean is, if someone is standing at one point on the globe, how far is it possible (in miles) to be away from that person, while still on the planet? And not at an uninhabitable place like the bottom of the Marianas Trench. Let's say I'm in Honolulu, Hawaii. What is the farthest place on the earth away from that point?

I'm going to take a stab at this using distance measurement.....20,039km or 12,451mi. Past these points, you would begin to get closer.
http://www.lyberty.com/encyc/articles/earth.html

12,450.775 miles

The circumference of the earth at the equator is 24,901.55 miles (40,075.16 kilometers).

http://geography.about.com/library/faq/blqzcircumference.htm

Skypilot024 is right, you would have to travel 12,541 miles before you get closer again. But seeing the earths circumference is longer round the equator than throungh to two poles, it would be logical to start at the city that lies on the equator, Quito in Equador. The furthest point from here using Google Earth is 12,453.8 miles and lies 21 miles south of Pekenbaru in Sumatra Indonisia. There may be some where slightly further but I haven't found it yet.