I assume expansion implies an increase in the volume of the universe. But if the volume of the universe is increasing, does that not imply the area it is expanding into's volume is being reduced?
In 1929, Hubble published his discovery that the sizes of galaxy redshifts and proportional to galaxy distances. That is, nearby galaxies are less redshifted than further away galaxies. This implies that further away galaxies are travelling away from us faster than nearby galaxies, which implies that we live in an expanding universe. We do not expand into anything, but rather, space is stretching. The Earth is not expanding, nor are you, me, or any other matter in the universe. It is space itself that is stretching.