Posts Tagged ‘isoperimetric’

Proof of the Isoperimetric Inequality 3

I will present here a third proof for the planar Isoperimetric Inequality, using some simple notions of differential curves. For this suppose that the simple closed plane curve {C} has length {L} and encloses area {A}. Then

\displaystyle L^2 \geq 4 \pi A

and equality holds if and only if {C} is a circle.

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Proof of the isoperimetric inequality 2

April 27, 2012 Leave a comment

I will continue the series of proofs for the isoperimetric inequality in the two dimensional case, i.e. if a simple closed curve { \Gamma} (which we suppose for simplicity that it consists of piecewise { C^1} curves) of length { L} encloses an area { A} then { L^2 \geq 4\pi A} and the equality is attained only if { \Gamma} is the boundary of a circle.

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Proof of the Isoperimetric Inequality

April 25, 2012 3 comments

The Isoperimetric inequality gives a bound for the area in terms of the perimeter of a set. It says that the greatest area that can be enclosed by a curve which has length L is maximal when the curve is the boundary of a circle, or equivalently the minimum of the perimeter of a curve which encloses a set of area A is attained again for the circle. In two dimension the inequality says that L^2 \geq 4\pi A where L is the perimeter and A is the area (Lebesgue measure) of a plane region \Omega.

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Region which can sustain the Largest Sandpile

April 24, 2012 Leave a comment

Among all plane regions \Omega which are open and simply connected and without holes of given area A the circle can support the largest sandpile.

Leavitt and Ungar – Circle supports the largest sandpile.

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Existence Result for the Isoperimetric Problems

November 5, 2011 5 comments

The tricky part is how to define the perimeter of a Lebesgue measurable set with finite perimeter. This can be done considering the space of bounded variation functions, denoted BV(\Bbb{R}^N). By definition we have for an open set U \subset \Bbb{R}^N that BV(U)=\left\{ f \in L^1(U) : \sup \left\{\int_U f {\rm div} \varphi dx | \varphi \in C_c^1(U;\Bbb{R}^N),\ |\varphi|\leq 1\right\} \right\}.  Here we denoted by C_c^1(U;\Bbb{R}^N) the space of continuously differentiable functions f : U \to \Bbb{R}^N with compact support in U. Because of the density of the space C_c^\infty(U,\Bbb{R}^N) of infinitely differentiable functions f: U \to \Bbb{R}^N with compact support in U in the space C_c^1(U;\Bbb{R}^N), we could have replaced C_c^1 by C_c^\infty in the above definition. You could take a look at this blog post for a detailed description of BV(U) or at the Wikipedia page.

We say that a set A of finite Lebesgue measure is a set of finite perimeter in \Bbb{R}^N if its characteristic function \chi_A belongs to BV(\Bbb{R}^N). This means that the distributional gradient \nabla \chi_A is a vector valued measure with finite total variation. The total variation |\nabla \chi_A| is called the perimeter of A.

In the same way we can define the perimeter of a Lebesgue measurable set A relative to an open set D. We say that A\subset D is a set of finite perimeter relative to D if the characteristic function \chi_A belongs to the space BV(D).

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