# Calculus of Variation

#### dwsmith

##### Well-known member
How do I set up the following problem?

What geometric surface encloses the maximum volume with the minimum surface area?

#### ThePerfectHacker

##### Well-known member
How do I set up the following problem?

What geometric surface encloses the maximum volume with the minimum surface area?
I think you want to rephrase it as, "What geometric surface encloses the maximum volume with a given surface area". In other words, you have 1 unit-squared of material and you want to enclose a surface to yields maximum volume.

Let $\Omega$ be bounded region in $\mathbb{R}^3$. You want to maximize,
$$\iiint_\Omega 1$$
Given the condition that,
$$\iint_{\partial \Omega} 1 ~ ds = 1$$

If we care about small details this question is a lot more complicated to phrase. Then we need to restrict ourselves to measurable sets such that .. blah blah blah.
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Here is another way to phrase it: Suppose $\Omega$ is a bounded (measurable) region in space with $\partial \Omega$ (boundary of $\Omega$) being a surface. Then,
$$\text{Volume}(\Omega) \leq \frac{1}{6\sqrt{\pi}} \text{Area}(\partial \Omega)^{3/2}$$

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#### dwsmith

##### Well-known member
I think you want to rephrase it as, "What geometric surface encloses the maximum volume with a given surface area". In other words, you have 1 unit-squared of material and you want to enclose a surface to yields maximum volume.

Let $\Omega$ be bounded region in $\mathbb{R}^3$. You want to maximize,
$$\iiint_\Omega 1$$
Given the condition that,
$$\iint_{\partial \Omega} 1 ~ ds = 1$$

If we care about small details this question is a lot more complicated to phrase. Then we need to restrict ourselves to measurable sets such that .. blah blah blah.
I don't think I want to rephrase it. The question is number 13 here

#### HallsofIvy

##### Well-known member
MHB Math Helper
That's still a very badly phrased question. I think they expect you to combine the facts that the smallest surface containing a given volume is a sphere and that the largest area for a given size surface is a sphere.

#### dwsmith

##### Well-known member
That's still a very badly phrased question. I think they expect you to combine the facts that the smallest surface containing a given volume is a sphere and that the largest area for a given size surface is a sphere.
Then how would I show a sphere is the smallest surface containing a given volume?

#### Deveno

##### Well-known member
MHB Math Scholar
This is a very complicated question without any "simple" answer, as the solution involves solving a non-linear partial differential equation of order 2.

The "best" answer that isn't too complicated I can think of is: soap bubbles are smarter than us, and they form spherical shells under such constraints.