Concerning sets of the first Baire category with respect to different metrics

• Autorzy: Moszyńska M. , Sójka G.
• Języki publikacji: EN

Abstrakty

EN

We prove that if ρH and δ are the Hausdorff metric and the radial metric on the space Sn of star bodies in R, with 0 in the kernel and with radial function positive and continuous, then a family A ⊂ Sn that is meager with respect to ρH need not be meager with respect to δ. Further, we show that both the family of fractal star bodies and its complement are dense in Sn with respect to δ.

Słowa kluczowe

EN

sets of the first Baire category; star body; fractal dimension; fractal star body

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2010
• Tom: Vol. 58, no 1
• Strony: 47--54
• Opis fizyczny: Bibliogr. 10 poz.

Twórcy

autor

Moszyńska M.

autor

Sójka G.

• Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland,

Bibliografia

• [1] W. L. Bloch, Fractal boundaries are not typical, Topology Appl. 154 (2007), 533-539.
• [2] K. Borsuk, Theory of Retracts, Polish Sci. Publ., 1967.
• [3] P. M. Gruber, In most cases approximation is irregular, Rend. Sem. Mat. Univ. Politec. Torino 41 (1983), 9-33.
• [4] P. M. Gruber, Dimension and structure of typical convex sets, continua and curves, Monatsh. Math. 108 (1989), 149-164.
• [5] P. Goodey and W. Weil, Intersection bodies and ellipsoids, Mathematika 42 (1995), 295-304.
• [6] I. Herburt and M. Moszyńska, On metric products, Colloq. Math. 62 (1991), 121-133.
• [7] I. Herburt, M. Moszyńska and D. Pronk, Fractal star bodies, in: Convex and Fractal Geometry, Banach Center Publ. 84, Inst. Math., Polish Acad. Sci., 2009, 149-171.
• [8] M. Moszyńska, Selected Topics in Convex Geometry, Birkhäuser, 2006.
• [9] R. Schneider, Convex Bodies: the Brunn-Minkowski Theory, Cambridge Univ. Press, 1993.
• [10] T. Zamfirescu, Description of most starshaped surfaces, Math. Proc. Cambridge Philos. Soc. 106 (1989), 245-251.