Geodesic
Widths related to pseudo-dimension
Izvestiya. Mathematics , Tome 73 (2009) no. 2, pp. 319-332

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We consider two widths related to the notion of pseudo-dimension. The first is $\rho_n$, which is defined in a similar way to Kolmogorov's width but replacing the linear dimension by the pseudo-dimension. $\rho_n$ can be bounded below by the second width $s_n$, which is half of the length of the maximal edge of the $(n+1)$-dimensional ‘coordinate’ cube inscribed in the given set in a special way. We construct examples of sets for which the ratios $\rho_n/s_n$ (for $n\geqslant 2$) and $\rho_{10n}/s_{9n}$ (for a sufficiently large $n$) are as large as desired. In terms of combinatorial dimension, the main result means that for any $C>0$ and any sufficiently large $n$ there is a set $W$ of dimension $\mathrm{vc}(W,1)\leqslant 9n$ which cannot be approximated with respect to the uniform norm with accuracy $C$ by any set $V$ of dimension $\mathrm{vc}(V,0)\leqslant 10n$.
Keywords: combinatorial dimension, widths.
Mots-clés : VC-dimension 
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     author = {Yu. V. Malykhin},
     title = {Widths related to pseudo-dimension},
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     number = {2},
     year = {2009},
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     url = {http://geodesic.mathdoc.fr/item/IM2_2009_73_2_a3/}
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Yu. V. Malykhin. Widths related to pseudo-dimension. Izvestiya. Mathematics , Tome 73 (2009) no. 2, pp. 319-332. http://geodesic.mathdoc.fr/item/IM2_2009_73_2_a3/