Well, first of all, you should state your question more precisely:
Alternative 1:
Of all geometric objects with the same volume, which has minimal surface area?
Alternative 2:
Of all geometric objects with the same surface area, which encloses the maximal volume?
Under certain assumptions of niceness, you may solve problems like these with the calculus of variations.
Yhen, in both cases, the ball (solid sphere) will be your solution.
