The geometric centre of Greater London is near the London Ambulance Service HQ in Waterloo, whilst weighted for population you end up in the Shell Centre on the South Bank. The centroid of the Congestion Charge Zone is on the other side of the river, near Somerset House, whilst according to Londonist the centroid of Zone 1 is on the Haymarket, near Piccadilly Circus. The statue of King Charles I seems a reasonable compromise between these four!