Concerning Sets of the First Baire Category with Respect to Different Metrics
Volume 58 / 2010
Bulletin Polish Acad. Sci. Math. 58 (2010), 47-54
MSC: 54E52, 52A30, 28A80.
DOI: 10.4064/ba58-1-6
Abstract
We prove that if $\varrho_H$ and $\delta$ are the Hausdorff metric and the radial metric on the space $\mathcal{S}^n$ of star bodies in $\mathbb R $, with 0 in the kernel and with radial function positive and continuous, then a family $\mathcal{A} \subset \mathcal{S}^n$ that is meager with respect to $\varrho_H$ need not be meager with respect to $\delta$. Further, we show that both the family of fractal star bodies and its complement are dense in $\mathcal{S}^n$ with respect to $\delta$.
