Posts Tagged ‘isoperimetric inequality’

Proof of the Isoperimetric Inequality 5

May 10, 2012 Leave a comment

Suppose {\Gamma} is a simple, sufficiently smooth closed curve in {\Bbb{R}^2}. If {L} is its length and {A} is the area of the region it encloses then the following inequality holds

\displaystyle 4 \pi A \leq L^2.

I am going to present another proof of the isoperimetric inequality. This time with Fourier series.

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Minkowski content and the Isoperimetric Inequality

May 9, 2012 5 comments

The proof of the isoperimetric inequality given below relies on the Brunn-Minkowski inequality and on the concept of Minkowski content.

In what follows we say that a curve parametrized by {z(t)=(x(t),y(t)),\ t \in [a,b]} is simple if {t\mapsto z(t)} is injective. It is a closed simple curve if {z(a)=z(b)} and {z} is injective on {[a,b)}. We say that a curve is quasi-simple if it the mapping {z} is injective with perhaps finitely many exceptions.

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The Brunn-Minkowski Inequality

May 9, 2012 1 comment

The Brunn-Minkowski Inequality gives an estimate for the measure of the set {A+B=\{a+b : a \in A,b \in B\}} in terms of the Lebesgue measures of {A} and {B} in {\Bbb{R}^d} (of course, we can only speak of such an inequality if all the sets {A,B,A+B} are Lebesgue measurable). The inequality has the form

\displaystyle |A+B|^{1/d} \geq |A|^{1/d}+|B|^{1/d}.

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Existence of Maximal Area Polygon with given sides

Suppose {a_1,..a_n,\ n \geq 3} are given positive real numbers such that there exists a polygon with {n} vertices having {a_1,..,a_n} as lengths of its size. Then among all polygons having these given side lengths there exists one which has maximal area.

Since I have used in the linked post the fact that the maximal area polygon with given sides exists, I will give a proof in the following.

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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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Cyclic polygon has largest area

April 28, 2012 Leave a comment

Suppose {a_1,..,a_n} are positive real numbers. Then the following statements are equivalent:

1. {a_1,...,a_n} satisfy the inequalities

\displaystyle 2a_i < a_1+...+a_n,\ i=1..n

2. There exists a polygon with sides {a_1,..,a_n}.

3. There exists a convex polygon with sides {a_1,..,a_n}.

4. There exists a cyclic polygon with sides {a_1,..,a_n}.

Of all these, the cyclic polygon has the maximal area.

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Isoperimetric inequality for polygons

April 27, 2012 3 comments

Among polygons with {n \geq 3} sides with given perimeter {P}, the one which maximizes the area is regular.

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