Shortest Path in a Polygon using Sublinear Space

$\renewcommand{\Re}{{\rm I!\hspace{-0.025em} R}} \newcommand{\SetX}{$\mathsf{X}$} \newcommand{\VorX}[1]{$\mathcal{V}$ \pth{#1}} \newcommand{\Polygon}{$\mathsf{P}$} \newcommand{\Space}{\overline{$\mathsf{m}$}} \newcommand{\pth}[2][!]{#1\left({#2}\right)}$ We resolve an open problem due to Tetsuo Asano, showing how to compute the shortest path in a polygon, given in a read only memory, using sublinear space and subquadratic time. Specifically, given a simple polygon $\text{Polygon}$ with $n$ vertices in a read only memory, and additional working memory of size $\text{Space}$, the new algorithm computes the shortest path (in $\text{Polygon}$) in $O(n^2\mathrm{/}\text{Space})$ expected time. This requires several new tools, which we believe to be of independent interest.