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Izv. RAN. Ser. Mat., 2010, Volume 74, Issue 1, Pages 135–158 (Mi izv2757)  

This article is cited in 1 scientific paper (total in 1 paper)

Widths of some classes of convex functions and bodies

V. N. Konovalova, V. E. Maiorovb

a Institute of Mathematics, Ukrainian National Academy of Sciences
b Department of Mathematics, Technion – Israel Institute of Technology, Haifa

Abstract: We consider classes of uniformly bounded convex functions defined on convex compact bodies in $\mathbb{R}^d$ and satisfying a Lipschitz condition and establish the exact orders of their Kolmogorov, entropy, and pseudo-dimension widths in the $L_1$-metric. We also introduce the notions of pseudo-dimension and pseudo-dimension widths for classes of sets and determine the exact orders of the entropy and pseudo-dimension widths of some classes of convex bodies in $\mathbb{R}^d$ relative to the pseudo-metric defined as the $d$-dimensional Lebesgue volume of the symmetric difference of two sets. We also find the exact orders of the entropy and pseudo-dimension widths of the corresponding classes of characteristic functions in $L_p$-spaces, $1\le p\le\infty$.

Keywords: convex function, entropy, pseudo-dimension.

DOI: https://doi.org/10.4213/im2757

Full text: PDF file (604 kB)
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English version:
Izvestiya: Mathematics, 2010, 74:1, 127–150

Bibliographic databases:

UDC: 517.5
MSC: 05B40, 41A10, 41A25, 41A45, 41A46, 42A61, 46A35, 60A05, 60C05, 60F15, 68T05, 68U05
Received: 10.01.2008
Revised: 29.12.2008

Citation: V. N. Konovalov, V. E. Maiorov, “Widths of some classes of convex functions and bodies”, Izv. RAN. Ser. Mat., 74:1 (2010), 135–158; Izv. Math., 74:1 (2010), 127–150

Citation in format AMSBIB
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\pages 135--158
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    This publication is cited in the following articles:
    1. S. V. Konyagin, A. A. Kuleshov, V. E. Maiorov, “Some problems in the theory of ridge functions”, Proc. Steklov Inst. Math., 301 (2018), 144–169  mathnet  crossref  crossref  isi  elib  elib
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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