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 $\gamma : [0, 1] \rightarrow \mathbb{R}^2$ has finite $p$-variation with $p < 2$, then $(\iint\limits_{\mathbb{R}^2}|\eta(\gamma, (x, y))|^q \,dx \,dy)^{1/q} \le (\frac{1}{2})^\frac{1}{q}(\zeta(\frac{2}{pq})-1)(||\gamma||_{p, [0, 1]})^{\frac{2}{q}} $ for all $q \in [1, \frac{2}{p})$, where $\eta(\gamma, (x, y))$ is the winding number of $\gamma$ at $(x, y), \zeta$ is the Reimann zeta function, and $||\gamma||_{p, [0, 1]}$ is the $p$-variation of $\gamma$ 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.