Concerning Sets of the First Baire Category with Respect to Different Metrics

Volume 58 / 2010

Maria Moszyńska, Grzegorz Sójka Bulletin Polish Acad. Sci. Math. 58 (2010), 47-54 MSC: 54E52, 52A30, 28A80. DOI: 10.4064/ba58-1-6


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 $\delta$.


  • Maria MoszyńskaInstitute of Mathematics
    University of Warsaw
    Banacha 2
    02-097 Warszawa, Poland
  • Grzegorz SójkaDepartment of Mathematics and Information Sciences
    Warsaw University of Technology
    Pl. Politechniki 1
    00-601 Warszawa, Poland

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