Near a Density Point Every Direction Meets the Set Closely

We work in the setting of Product Measures and Lebesgue Measure on ℝⁿ: Standing Notation and Background, whose notation is fixed for every dimension and is used here with a natural number $n$ satisfying $1\le n$: the Euclidean norm $\lVert\,\cdot\,\rVert$, distance $d_{E}$ and closed balls $\bar{B}(x,r)$ of $\mathbb{R}^{n}$, the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R}^{n})$, and Lebesgue measure $\lambda_{n}$ are as fixed there. Write $\lambda_{n}^{\ast}$ for Lebesgue outer measure, and let density point have the meaning fixed in that definition.

Let $G\subseteq\mathbb{R}^{n}$ be an arbitrary subset, let $x\in\mathbb{R}^{n}$ be a density point of $G$, and let $\delta\in\mathbb{R}$ with $0<\delta$. Then the following holds.

1. (A nearby point of the set) (i) There is $\rho\in\mathbb{R}$ with $0<\rho$ such that for every $w\in\mathbb{R}^{n}$ with $0<\lVert w-x\rVert<\rho$ there is $g\in G$ with

$$\lVert g-w\rVert\le\delta\,\lVert w-x\rVert .$$