In a 3-dimension isoperimetric problem, a ball maximizes the volume

In summary, the conversation discusses the difficulty of using functional derivative in R^3, specifically in solving the isoperimetric problem. The use of Lagrange multipliers is mentioned, but it is noted that it can only solve the problem in R^2. The conversation then shifts to finding a proof for the isoperimetric problem in R^3, with the realization that it is a difficult task. The suggestion of using the Brunn-Minkowski theorem is mentioned as a potential solution.
  • #1
graphking
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TL;DR Summary
isoperimetric problem: in R^n, fix the surface area, when the volume can be max? the answer is ball, like B(0,1). in R^2 you can find many ways, such as using the variation of fixed end curve, functional derivative. But in R^3 I found it hard to use functional derivative (the equation get from derivative=0 is complicated, I can't further get to B(0,1))
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  • #2
graphking said:
But in R^3 I found it hard to use functional derivative (the equation get from derivative=0 is complicated, I can't further get to B(0,1)
I suppose you are referring to Lagrange multipliers ?
If so, what's the problem ?

If not, please post your work

##\ ##
 
  • #4
BvU said:
I suppose you are referring to Lagrange multipliers ?
If so, what's the problem ?

If not, please post your work

##\ ##
Using the lagrange multiplier is a way can only solve the isoperimetric problem in R^2, I show you the result you get in R^3:
assuming the surface is z(x,y), with fixed boundary in XOY plane, then
##(z_x/(1+z^2_x+z^2_y)^{1/2})_x+(z_y/(1+z^2_x+z^2_y)^{1/2})_y \equiv -1/\lambda
##
 
  • #5
Is there a question here? If i understand you correctly, you tried to use the ideas of one of the solutions to the problem in dimension two to solved it in higher dimensions, but you couldn't. So? The problem is not easy. May be this approach doesn't generalize or may be it does and you couldn't do it. Are you asking how it is done?
 
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  • #6
martinbn said:
Is there a question here? If i understand you correctly, you tried to use the ideas of one of the solutions to the problem in dimension two to solved it in higher dimensions, but you couldn't. So? The problem is not easy. May be this approach doesn't generalize or may be it does and you couldn't do it. Are you asking how it is done?
please teach me a good way to proof the isoperimetric problem in R^3
 

FAQ: In a 3-dimension isoperimetric problem, a ball maximizes the volume

What is the isoperimetric problem in three dimensions?

The isoperimetric problem in three dimensions involves determining the shape that encloses the maximum volume for a given surface area. In simpler terms, it seeks to find the relationship between the volume of a solid and its surface area, specifically identifying which geometric shape can maximize volume while minimizing the surface area.

Why does a ball maximize the volume in the isoperimetric problem?

A ball, or sphere, maximizes volume for a given surface area because of its symmetrical properties and uniform distribution of points. The sphere has the least surface area relative to its volume compared to any other three-dimensional shape, which leads to the conclusion that it encloses the maximum volume for a fixed surface area.

How is the isoperimetric inequality related to the isoperimetric problem?

The isoperimetric inequality is a mathematical statement that formalizes the relationship between surface area and volume. It states that for any closed surface in three-dimensional space, the ratio of the surface area squared to the volume is minimized by the sphere. This inequality provides a foundation for understanding why the sphere is the optimal solution to the isoperimetric problem.

Are there other shapes that can maximize volume in specific conditions?

What are some applications of the isoperimetric problem in real life?

The isoperimetric problem has several practical applications, including in fields like biology, architecture, and materials science. For instance, it can help in understanding the shapes of cells and bubbles, designing efficient packaging, optimizing the use of materials in construction, and even in minimizing the energy required for certain physical processes. The principles derived from the isoperimetric problem aid in creating solutions that maximize efficiency and minimize waste.

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