With 3 hot dog stands in 3 dimensions they would still be very close together. This trivially holds true as we are living in an N dimensional space and there are N or less stores.
I think a more generalized solution will still work with this problem. When there's no center (ie beach extends infinitely), the hot dog vendors would choose to be close together for the simple fact that any distance between the two they share equally. Both vendors would seek to minimize this because any distance outside they claim completely for themselves.