Theorem isoperimetric

The theorem isoperimetric is a result of extremality relating to the surface of a field enclosed by a closed curve. The Cercle is the closed Courbe smaller length enclosing a related field of surface given. Known as differently, the circle is the closed curve enclosing a related field of maximum surface for a given length. This fact knew its first true demonstration only in 1875 by Schwarz.

One of the methods of proof, known since the demonstration of Hurwitz in 1901 is to use a result of analysis, resulting from the theory of the Fourier series, known under the name of Inégalité of Wirtinger.

Statement of the theorem

That is to say a closed Curve defined by a function F (T) = (X (T), there (T)) periodic, continuously derivable. Are L its length and has the algebraic surface field which it limits. Then

$L^2\; \backslash \; geq\; 4\; \backslash \; pi\; A$ Moreover there is equality if and only if the curve is a circle.

Finally the theorem remains true when one supposes the curve only class $\backslash \; mathcal\; C^1$ per pieces and continuous.

Recall of the inequality of Wirtinger

That is to say F a function 2π - periodic, of average null, class $\backslash \; mathcal\; C^1$ per pieces and continuous. Then

$\backslash |F\; \backslash |^2\; =\; \backslash \; frac1\; \{2\; \backslash \; pi\}\; \backslash \; int\_\; \{0\}\; ^\; \{2\; \backslash \; pi\}\; (F\; (T))^2\; dt\; \backslash \; Leq$

\|f' \|^2 = \ frac1 {2 \ pi} \ int_ {0} ^ {2 \ pi} (f' (T))^2 dt~

Moreover if ||F||=||f'||, then F is a sinusoidal function

$f\; (X)\; =A\; \backslash \; cos\; (X)\; +B\; \backslash \; sin\; (X)\; =C\; \backslash \; cos\; (x+\; \backslash \; varphi)$

One will find the demonstration in the article Inégalité of Wirtinger.

Demonstration

That is to say T the period of the curve. Let us point out the formulas first of all expressing L and has

$L=\; \backslash \; int\_0^T\; \backslash \; sqrt\; \{x\text{'}\; (T)\; ^2+y\text{'}\; (T)\; ^2\}\; D\; T\; \backslash \; qquad\; has\; =\; \backslash \; int\_0^T\; X\; (T)\; y\text{'}\; (T)\; D\; t$

One carries out a certain number of simplifications

• change of parameter (what does not change L nor has )

To get rid of the square root in the expression length, one uses like parameter the curvilinear Abscisse S , i.e. the arc is traversed at uniform speed 1. Then L is also equal to the energy E defined by

$E=\; \backslash \; int\_0^T\; (x\text{'}\; (S)\; ^2+y\text{'}\; (S)\; ^2)\; D\; S\; =\; L\; =\; T$

• Homothétie (what changes L and has without modifying the $L^2/A$ report/ratio)

One brings back oneself by homothety to an arc length L=2π=T .

• translation (what does not change L nor has )

One brings back oneself to X of null median value.

One is finally ready to calculate

$L^2-4\; \backslash \; pi\; has\; =\; 2\; \backslash \; pi\; (L-2A)\; =2\; \backslash \; pi\; \backslash \; int\_0^\; \{2\; \backslash \; pi\}\; \backslash \; left\; (x\text{'}\; (S)\; ^2+y\text{'}\; (S)\; ^2-2x\; (S)\; y\text{'}\; (S)\; \backslash \; right)$

ds

As X checks the assumptions of the inequality of Wirtinger:

$L^2-4\; \backslash \; pi\; has\; \backslash \; geq\; 2\; \backslash \; pi\; \backslash \; int\_0^\; \{2\; \backslash \; pi\}\; \backslash \; left\; (X\; (S)\; there\; \text{'}\; (S)\; \backslash \; right)\; ^2$

ds\geq 0

In the case of equality: on the one hand it is necessary that there is equality in the inequality of Wirtinger, which gives the expression of X (S) . In addition so that this integral is cancelled one must have y' (S) =x (S) . These two conditions give well a circle of radius 1, parameterized at uniform speed. But of course all the circles are solutions, whatever the way of parameterizing them (it is necessary to apply in opposite direction the simplifying assumptions).

In dimension 3

The theorem isoperimetric admits a generalization in dimension 3.

The theorem of Jordan affirms that the curves of Jordan separate the plan in two related components. In dimension higher, all hypersurface related is directional and borders a single related field.

The spheres are compact surfaces related which with fixed surface maximize the interior volume, or which, with fixed interior volume, minimize surface.

Thus the soap bubbles which the children make are indeed spheres!

See too

• isoperimetric Inequality

References

• Mr. BERGER and R. GOSTIAUX, differential Geometry, P.U.F., Paris 1986.

• Y.D Burago, V.A. Zallgaller, Geometric inequalities, Springer Grundlehren Maths. 285, Springer 1988

Isopérimétrique

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