Recent content by eee000

Thank you haruspex! That's what I was looking for. I was thinking in more local terms, along the lines of Steiner's arguments, but your one-liner is much better.

For the partial sphere, the solution is easy - the half sphere is optimal. The open problem (to the extent of my minimal knowledge) is: "For an open manifold of unit volume(*) , what shape will yield minimal surface(*) area?" where Volume is the volume contained up to the first hole, and...

sure it is, but I think the correct formulation of the problem is that of a fixed volume, especially if one is interested in he general question - what is the best shape of all. Since the A/V ratio generally scales with 1/V^(1/3), comparison must be made for equal volumes.

I'm sorry,I made a mistake too... Since the surface to volume ratio scales as 1/V^(1/3) we need to do the optimization for fixed volume. This yields that the optimal shape is exactly the half sphere. Obviously, my statement "(64 Pi/3)^(1/3), better than the half sphere with (18 Pi)^(1/3)"...

Thank you for the spherical cap formulae, but I think your algebra is wrong. Fixing the sphere radius R, and parameterizing the cap by h, I find the volume to be V=Pi h^2 (R - h/3) and the surface S=2 Pi R h. The lowest ratio is obtained when h=3R/2, and then V=9Pi/8 R^3, and S=3Pi R^2. Setting...

Thank you, The top I have in mind is defined by gravity. I want to have a vessel full of water that do not spill out. That is, if we define the z axis to be the direction of gravity, I think of a surface that may have holes in it, but the volume that we count is only integrated up to lowest z...

Hi, It is well known that a sphere has the lowest surface to volume ratio. However, a related question is: What is the shape that gives the lowest surface to volume ratio if you do not include the "top" in the surface. That is, what is the maximal volume of an uncovered vessel of a fixed...