Let's first take the case of a beam supported on two points. The easiest way of reasoning is dividing the weight of the beam equally over the 2 points. This is the same as cutting the beam in two equal pieces and balancing each of them on one point. Balancing means supporting the half beam in the middle.For further comparison we speak of supporting on the 50% zone.
However in real life, the beam is not cut in two, and in that case supporting the beam on the same points the deflection in the middle is smaller than on the tops ! This is due to the fact that when the beam is cut in two parts, the end-faces will not be anymore vertical, but rotated. The same faces of the whole beam are connected with each other and stay vertical, so curving upward and deflecting so much. Handbooks on measurement techniques learn us that by supporting the beam on the 55% zone, the deflections of the ends and in the middle will be the same, creating an optimum support radius for minimal deflection.
We learn from this simple case, that the equal weight principle is a approximation of
the real case and that the real optimimum radius is somewhat more outwards. If we now
consider a flat disk: equal weight principle means supporting at 70.7% radius. But in
analogy with the beam, the middle has more strenght because it's connected to the other
pieces of pie !
So the supporting zone to have minimal defeflection for a flat disk will also be somewhat further outwards. Using equal weight principle means using radii who are to small giving the flat disk a convex shape.
However there are two remarks : a telescope can be refocused, so deflection can be counterd more or less by using the nearest parabolo´dal form. The disk in a telescope is not flat, but has a hollow front-side and some are perforated or have a shell-form. So the problem is complexer than at first thought. On this site we consider only parabo´dal mirrors (the possibility of central hole is mentioned when appropiate) and a flat back side and refocussing is mentioned when used. In fact supporting a thin flat, circular plate on a continuous ring (shear forces not taken into account) should be done on 68.1% if it should stay as much as possible flat. If defocus is tolerated you can have it on 57.3%.
A paper of Nelson et al. points out that more than 6 points is indistinguishable from 6 points. So the figures above do stay the same for down to 6 points. The case of 4 points if defocus is not tolerated is about 67% (taken from a graph). I did not find values for the case if defocus is tolerated.
Last but not least the 3 point support should be on 64.5% if defocus not tolerated and on 39.9% if defocus is tolerated according to a paper of Luc Arnold. Toshimi Taki did point out the same values independently.
Let me mention again that this figures are for thin disks and shear is not taken into account.
For the real purists : Luc Arnold is given also graphs where the central obscuration is taken into account, but for normal amateur telescopes it's all withing the range of a few percent, except for the case of the 3 point support with tolerated focus. If there's 0.3 obscuration one can leave it on 35%, for 0.2 38%. 0.1 and 0 obscuration is pratically the same.
About the shear force : Richard Schwartz as Luc Arnold have pointed out independently that by using support pads instead of points the peak-to-valley error is noticeably reduced in the neighbourhood of the pads, but RMS remains alomost the same. Shear force comes mostly into the game when the thickness is not neglectable in comparison with the distance between two points, which is not the case here.