6533b81ffe1ef96bd127846d

RESEARCH PRODUCT

On the conical density properties of measures on $\mathbb{R}^n$

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We compare conical density properties and spherical density properties for general Borel measures on $\mathbb{R}^n$ . As a consequence, we obtain results for packing and Hausdorff measures $\mathcal{P}_h$ and $\mathcal{H}_h$ provided that the gauge function $h$ satisfies certain conditions. One consequence of our general results is the following: let $m, n\,{\in}\,\mathbb{N}, 0\,{\lt}\,s\,{\lt}\,m\,{\leq}\,n$ , $0\,{\lt}\,\eta\,{\lt}\,1$ , and suppose that $V$ is an $m$ -dimensional linear subspace of $\mathbb{R}^n$ . Let $\mu$ be either the $s$ -dimensional Hausdorff measure or the $s$ -dimensional packing measure restricted to a set $A$ with $\mu(A)\,{\lt}\,\infty$ . Then for $\mu$ -almost every $x\,{\in}\,\mathbb{R}^n$ , there is $\theta\,{\in}\,V\,{\cap}\, S^{n-1}$ such that \[ \liminf\limits_{r\downarrow 0}r^{-s}\mu(B(x, r)\cap H(x,\theta,\eta))=0,\] where $H\left(x,\theta,\eta\right)\,{=}\,\{{y\in\mathbb{R}^n}{\ip{\br{y-x}}{\theta}\,{\gt}\,\eta|{y-x}|\}$ .