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Widths of some classes of convex functions and bodies. (English. Russian original) Zbl 1196.41017
Izv. Math. 74, No. 1, 127-150 (2010); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 74, No. 1, 135-158 (2010).
The abstract to be recalled tells clearly the subject and the results of this essential work.
“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, $$1\leqslant p\leqslant \infty$$.”

##### MSC:
 41A46 Approximation by arbitrary nonlinear expressions; widths and entropy 41A25 Rate of convergence, degree of approximation
##### Keywords:
convex function; entropy; pseudo-dimension
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