Exercise 44, § 1

Let $I = [a, b]$ be a compact interval of $\mathbf{R}$. Let $f$ be a continuous function from I into $\mathbf{R}^2$ which is injective on the interior of I, and which is piecewise $C^1$ (that is to say, there exists a finite partition of I into intervals J such that $f|J$ is differentiable, and $f(f|J)'$ is continuous). One says that $f$ is a parametrized curve in the plane. One says that

(where $f'(t)$ is identified with a complex number) is the length of the parametrized curve $f$. If moreover $f(a) =f(b)$, one says that $f$ defines a Jordan curve. There then exists a unique bounded connected component C of $\mathbf{R}^2-f(I)$ (TA, to appear). The Lebesgue measure of C is called the area enclosed by the Jordan curve $f$, and is denoted by $A(f)$.

(a) Suppose that $f$ is a Jordan curve. For every $t\in I$, the restriction $f_t$ of $f$ to $[a, t]$ is a parametrized curve and the function $s:t\mapsto \ell (f_t)$ is a strictly increasing bijection from I onto $J = [0, \ell (f)]$. The function $g=$ $f\circ s^{-1}: J\rightarrow \mathbf{R}^2$ is a Jordan curve, and for every $t\in J$, one has $\ell (g_t) =t$. (b) Let $f= (f_1, f_2)$ be a Jordan curve. One has

2 $_I$

where the integral is with respect to the Lebesgue measure on I. (Apply the Stokes formula.) Moreover, one has $A(g) = A(f)$, where $g$ is the function of question a).

(c) One has $4\pi A(f)\leqslant \ell (f)^2$ (“isoperimetric inequality in the plane”). (Reduce to the case where $\ell (f) = 1$; consider the Fourier series of the real and imaginary parts of $g$ and now prove that $|g'(t)|^2= 1$ for $t\in J$.)

(d) If $4\pi A(f) =\ell (f)^2$, then the image of $f$ is a circle in $\mathbf{R}^2$.