Proving the Sphere Maximizes Volume Among All Simple Closed Surfaces with a Given Surface Area

To prove that a sphere maximizes the volume enclosed among all simple closed surfaces of a given surface area, we can use the isoperimetric inequality. This fundamental result in mathematics relates the surface area and volume of a shape, leading us to this fascinating geometric analysis.

Steps of the Proof

Isoperimetric Inequality

The isoperimetric inequality states that for any simple closed curve C in R3 with length L and the volume V of the region R it encloses, the following inequality holds:

V ≤ L3/36π

Equality is achieved if and only if R is a sphere.

Surface Area and Volume Relation

For a surface with a fixed surface area S, we want to maximize the volume V. The surface area S of a sphere of radius r is given by:

S 4πr2

The volume V of the sphere is given by:

V 4/3πr3

Setting Up the Problem

Given a fixed surface area S, we can express r in terms of S:

r (S/4π)1/2

Substituting this expression for r into the volume formula gives:

V 4/3π · (S3/2 / (4π3/2) S3/2 / (6√π)

Maximization

The volume V as a function of S is:

V(S) S3/2 / (6√π)

This function is increasing for S 0. Thus, for a fixed surface area S, the volume V is maximized when the shape is a sphere.

Conclusion

Since a sphere achieves the equality in the isoperimetric inequality, we conclude that among all simple closed surfaces with a given surface area, the sphere encloses the maximum volume.

Summary

The sphere maximizes the enclosed volume for a given surface area due to the isoperimetric inequality, which states that for any closed surface, the volume will always be less than or equal to that of a sphere with the same surface area. Thus, the sphere is the optimal shape for maximizing volume given a fixed surface area.

Complementary to Minimizing Surface Area for a Given Volume

This analysis is complementary to minimizing the surface area for a given volume. If you have two simple closed surfaces enclosing the same volume, one of them is a sphere, the sphere will have the smaller surface area. When both surfaces are scaled by the same factor required to make their surface areas equal, the sphere will be scaled up the most, enclosing a greater volume than the other surface. The converse is equally easy to prove.

Optimality in Both Questions

The sphere is optimal for both maximizing the enclosed volume and minimizing the surface area for a given volume. This is not straightforward and can be shown using an intuitive approach involving convexity and symmetry.

Intuitive Approach

First, take any plane through the centroid of the solid. If one side has a greater surface area, replace it with a reflection of the other half. This means the optimal solid could have mirror symmetry about any plane through the centroid. This inequality does not show that mirror symmetry is necessary. Instead, consider constructing a symmetrical solid with the same optimal property.

The optimal solid must be convex. If not, find a line that meets the solid at a point of concavity. There is a plane perpendicular to that line that meets the surface in a closed curve. Replace part of the surface by part of that plane, simultaneously increasing the volume and decreasing the surface.

We have already established that an optimal solid is convex and symmetric. Furthermore, tangents to the surface must be perpendicular to radii measured from the centroid. If not, reflecting the surface in a plane containing the radius will produce a non-convex surface. This method doesn't apply at points where the surface is not smooth.

If the radius measured from the centroid is not constant, there is at least one pair of antipodal points with the maximum radius.

While these steps provide an intuitive understanding, a rigorous proof may require more advanced mathematical techniques, including calculus of variations and Fourier analysis.

These two problems are deeply interconnected and illustrate the fundamental principles of optimization in geometry, relating surface area and volume.