Areas spanned by point configurations in the plane

Abstract

We consider an over-determined Falconer type problem on $(k+1)$-point configurations in the plane using the group action framework introduced in \cite{GroupAction}. We define the area type of a $(k+1)$-point configuration in the plane to be the vector in $\R^{\binom{k+1}{2}}$ with entries given by the areas of parallelograms spanned by each pair of points in the configuration. We show that the space of all area types is $2k-1$ dimensional, and prove that a compact set $E\subset\R^d$ of sufficiently large Hausdorff dimension determines a positve measure set of area types.