Geodesic
Widths of some classes of convex functions and bodies
Izvestiya. Mathematics , Tome 74 (2010) no. 1, pp. 127-150

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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
Mots-clés : pseudo-dimension. 
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     author = {V. N. Konovalov and V. E. Maiorov},
     title = {Widths of some classes of convex functions and bodies},
     journal = {Izvestiya. Mathematics },
     pages = {127--150},
     publisher = {mathdoc},
     volume = {74},
     number = {1},
     year = {2010},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2010_74_1_a1/}
}
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V. N. Konovalov; V. E. Maiorov. Widths of some classes of convex functions and bodies. Izvestiya. Mathematics , Tome 74 (2010) no. 1, pp. 127-150. http://geodesic.mathdoc.fr/item/IM2_2010_74_1_a1/