Optimum shapes of reservoirs

The aim of this paper is to find the optimal shape of reservoirs and to minimize the whole lateral surface area for a known volume V. The minimization is studied for some geometrical shapes generated by rotation. It is proved that the lateral surface area of a hollow ellipsoid with negligible or non negligible thickness is minimized when it is transformed to a sphere and a formula is proved for the calculation of lateral surface area of ellipsoid. It is then proved that the minimization of the whole lateral surface area with a constant volume results in a differential equation which cannot be resolved in general case; but only for some particular boundary values.