This will be a cool application of the concepts introduced in the last post. We call a curve **rectifiable** if there exists finite such that for any partition , , i.e. the curve has some notion of “finite length,” where length is the supremum of this quantity taken over all partitions or equivalently the infimum over all upper bounds . It follows by definition that if the – and -parametrizing functions of the curve are of bounded variation, the curve is rectifiable. It is not, however, true that the classical arc length formula holds for all curves of bounded variation: just consider a curve where the and functions are the Cantor-Lebesgue function. We get a straight line from the origin to , but the derivative of the function is zero almost everywhere.

Result 4: The arc length formula does work if we assume absolute continuity of and .

Proof: We will prove that the total variation of a complex valued function over is , because then we can just substitute . As usual, we will prove that there is inequality in both directions. But recall that for absolutely continuous functions, the fundamental theorem of calculus holds, so pick a partition so that . In the other direction, recall that step functions are dense in , so we can find an approximating step function to so that has integral arbitrarily small. By the triangle inequality, , where and . We can bound by taking a partition where the adjacent intervals are over constant parts of the step function so that . But recall that we picked to be extremely close to so that , so and we get inequality in the other direction.

Now before we proceed to state and prove the isoperimetric inequality, let’s get some vocabulary under our belt. Define the one-dimensional **Minkowski content** of a curve to be , where denotes the set of points which are at most away from any point in . Define a **simple curve** to be a curve that doesn’t double over on itself, a **quasi-simple** curve to be a curve such that is injective except for finitely many points, and a **closed curve** to be one that starts where it ends. As the name suggests, the Minkowski content of a curve turns out to be precisely its length if the curve is rectifiable and quasi-simple.