An elementary proof of an isoperimetric inequality for paths with finite $p$-variation

Abstract

In this article we will prove that if the continuous closed curve $γ: [0, 1] \rightarrow \mathbb{R}^2$ has finite $p$-variation with $p < 2$, then $(\iint\limits_{\mathbb{R}^2}|η(γ, (x, y))|^q \,dx \,dy)^{1/q} \le (\frac{1}{2})^\frac{1}{q}(ζ(\frac{2}{pq})-1)(||γ||_{p, [0, 1]})^{\frac{2}{q}} $ for all $q \in [1, \frac{2}{p})$, where $η(γ, (x, y))$ is the winding number of $γ$ at $(x, y), ζ$ is the Reimann zeta function, and $||γ||_{p, [0, 1]}$ is the $p$-variation of $γ$ on the interval $[0, 1]$. Our main contribution is that we have explicitly given a bound by known constants, and we have found this by an elementary proof. We are going to be using a method introduced by L.C. Young in 1936.