A Marstrand theorem for measures with polytope density

Abstract

Given \({s\in (0,2]}\) and any centrally symmetric convex polytope \({\Theta\subset\mathbb{R}^{n}}\) , define \({\Theta_r(x):=r\Theta+x}\) we prove that if a Radon measure μ has the property \({\label{df1} 0 < \mathop {\lim }\limits_{r \to 0} \frac{\mu\big(\Theta_r(x)\big)}{r^s} < \infty\quad {\rm for}\;\mu\textrm{ a.e. }x}\) then s is an integer. For the case Θ is the Euclidean ball, this result was first proved by Marstrand in 1955 for Hausdorff measure in the plane (Marstrand in Proc Lond Math Soc 3(4):257–302, 1954) and later for general Radon measures in \({\mathbb {R}^n}\) (Marstrand in Trans Am Math Soc 205:369–392, 1964).