pace is one of the most important elements in chess in general. Pawn endgames are no exception. One of the conditions for a successful breakthrough is the advantage in territory.
However pure space advantage means more advanced and better enrgized pawns and might also be a key factor in the endgame:

[It seems as White is losing as the enemy king is more active. However, there is a deeper detail which decides the battle- space!]

1.Kc3! [White needs to win the opposition if they want to win!]

[1.Kd3? Kd5=]

1...Kd5 [1...Kf5 2.Kc4 Kg5 3.Kc5 Kxh5 4.Kb6 Kg4 5.Kxa6 h5 6.Kb6 h4 7.a6 h3 8.a7 h2 9.a8Q+–; 1...Ke6 2.Kc4 is winning for White, too.]

2.Kd3 Kc5 3.Ke4 Kb5 4.Kf5 Kxa5 5.Kg6 Kb4 6.Kxh6 a5 7.Kg6 a4 8.h6 a3 9.h7 a2 10.h8Q

Just in time! All of this was possible thanks to the space advantage of the white pawns!
The corner pawns proved faster thanks to their better energy and together with the geometrical motifs gave the win to white.
Surprisingly, a few days ago the same position could have occurred in the match between the "Snowdrops" and "Oldhands":
Uhlmann,Wolfgang (2319) - Tania,Sachdev (2400)
Snowdrops vs Oldhands Podebrady CZE (5.3), 13.12.2012

47.e3?? [One move too short. The legendary GM could have won with:]

[47.e4! dxe3 48.fxe3 Kf6 49.e4 fxe4 50.dxe4 c4 51.e5+! The pawns are swapped off and after: 51...Ke6 52.Ke4 c3 53.Kd3 Kxe5 54.Kxc3+– We suddenly reach the position from the Grigoriev's study! 54...Kd5 55.Kd3 Kc5 56.Ke4 Kb5 57.Kf5 Kxa5 58.Kg6 Kb4 59.Kxh6 a5 60.Kg6 a4 61.h6 a3 62.h7 a2 63.h8Q (a line given by chesstoday.com)]

47...c4!–+ [Breakthrough. It is now Black who suddenly wins!]

48.dxc4 d3 49.Kf3 Ke5

Chess is a cruel game sometimes...