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 surface area? For example, a cylinder whose height is equal to its radius, has a volume Pi R^2 h = Pi R^3 and a surface (without cover) of Pi R^2 + 2*Pi *R*h = 3 Pi R^2. If we fix the volume to unity, the surface in this case is 3.Pi^(1/3). In comparison, a half sphere of unity volume has a smaller surface -- (18 Pi)^(1/3). However, it's clear this is not the best shape - a sphere cut a bit above half its volume beats the half sphere. So, what is the shape that gives the lowest surface to volume ratio if you do not include the "top" in the surface ? Thanks!