I am trying to find an optimum shape for a clamshell container (one that has the least surface area to volume ratio). While this should be a sphere, a spherical container is not feasible as its hard for the container to stay in one position while eating from it. Hence, the shape I proposed was a partially spherical container with a flat base. However, I am unsure as to minimise the surface area to volume ratio of this shape (I know this) while at the same time maximizing its stability. This is I don't know how to model stability, which varies with height of the container and base area of the container. Hence, I would like to ask if it is even possible to achieve this.
Thanks everybody!
Calculate the center of mass of the container and typical contents (usually modeling the container as a fixed-density material, and contents as water, suffices).
Then, calculate the solid angle where the center of mass is supported by the base of the container.
The larger the solid angle, the more stable it is: the more you can tilt it before it topples. (Toppling over occurs when the center of mass is not supported by the base of the object.)
You can also "cheat" by using dissimilar materials. For example, a very lightweight (but food-safe!) plastic for the container itself, but with a chunk of tungsten (or lead, although I really don't like the idea of having lead in my food containers) in the base. This moves the actual center of mass very low, very close to the base; therefore the solid angle the base covers from the center of mass will be high, and thus the container very stable.
In your case, you have two qualities you want to optimize at the same time: minimize the surface area to volume ratio, and maximize the solid angle the base covers from the center of mass of a container (with various amounts of contents of various density distributions, too). Mathematically, the really hard part is to choose the criterion on how these qualities are combined to yield the final "desirability" value (so that we can optimize it).
In reality, there are additional restrictions to the optimum solution. For example, if the only opening in the container is a small hole, it is difficult to put anything non-liquid in it, or use it with eating utensils. When considering the stability of the container, the opening should also be considered: a container that will spill its contents easily, but then revert back to its stable position, is almost as annoying as one that ends up being tipped over.
I'd say that it is much harder to design the actual mathematical statements, than solving them. Many designers avoid the mathematical step, by simply constructing several prototypes, and testing them. (Essentially, that way you let reality handle all the aspects math for you. Otherwise it is very much a similar process.)
You can easily stabilize a spherical container by removing a thin spherical cap, and/or adding three stabilizing feet (as Vasya mentioned in a comment to the question). Neither changes the volume or the surface area much, but really affect the stability of the container a lot. (In particular, many containers have a ring-shaped "foot" at the bottom. The volume of this support ring is not that big, but it really makes many shapes quite stable.)
However, spherical containers with a small opening are not easy to eat from, nor to clean. (In particular, many dish washing machines may not get the inner upper surfaces clean, because the jets of water never hit those areas directly.) My big (about one liter) soup/ramen/hot cocoa ceramic bowl I use when I have a cold (so it acts partially as a hot bottle, and does not get cold too fast) is somewhat spherical, with the opening diameter about 3/4 of the maximum diameter. So, even nice spherical bowls are not full spheres. (The opening is a bit flared out for stopping drippage, and it even has two tiny ears to move it around when the sides are a bit too hot to handle. Those old-timey pot/bowl makers actually optimized these shapes quite well over the centuries, in my opinion.)
(As I wrote this, I realized that particular bowl keeps warm so long, especially if I use a platter as a lid, because of both the material and the semi-spherical shape. Less surface area, less heat convection.)
Okay, now I got a hankering for nice hot chicken soup. Grr.