It's intuitive to me that for a given volume, a sphere will have the smallest surface area to volume ratio. But how do we prove it?
1) Since the volume is fixed, there is no need to theorize on the ratio. Examining the Surface Area will be sufficient.
2) Have you considered the examination of one dimension less? For a given area, a circle presents the least circumference.
3) You may need to experiment with different shapes.
1 ) "Calculus of variations"
2) ?? You don't know "the spherical shape isn't a "local minimum"; it is!