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Izv. RAN. Ser. Mat., 2009, Volume 73, Issue 2, Pages 109–122 (Mi izv2648)  

Widths related to pseudo-dimension

Yu. V. Malykhin

M. V. Lomonosov Moscow State University

Abstract: 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 $n\geq 2$) 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 $\mathrm{vc}(W,1)\leq 9n$ which cannot be approximated with respect to the uniform norm with accuracy $C$ by any set $V$ of dimension $\mathrm{vc}(V,0)\leq 10n$.

Keywords: VC-dimension, combinatorial dimension, widths.

DOI: https://doi.org/10.4213/im2648

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English version:
Izvestiya: Mathematics, 2009, 73:2, 319–332

Bibliographic databases:

UDC: 519.1+517.5
MSC: Primary 41A46; Secondary 54F45
Received: 11.04.2007

Citation: Yu. V. Malykhin, “Widths related to pseudo-dimension”, Izv. RAN. Ser. Mat., 73:2 (2009), 109–122; Izv. Math., 73:2 (2009), 319–332

Citation in format AMSBIB
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