Geodesic
The Product of Octahedra is Badly Approximated in the $\ell_{2,1}$-Metric
Matematičeskie zametki, Tome 101 (2017) no. 1, pp. 85-90.

Voir la notice de l'article provenant de la source Math-Net.Ru

We prove that the Cartesian product of octahedra $B_{1,\infty}^{n,m}=B_1^n\times \dots\times B_1^n$ ($m$ factors) is poorly approximated by spaces of half dimension in the mixed norm: $d_{N/2}(B_{1,\infty}^{n,m},\ell_{2,1}^{n,m})\ge cm$, $N=mn$. As a corollary, we find the order of linear widths of the Hölder–Nikolskii classes $H^r_p(\mathbb T^d)$ in the metric of $L_q$ in certain domains of variation of the parameters $(p,q)$.
Keywords: Kolmogorov width, vector balancing. 
@article{MZM_2017_101_1_a6,
     author = {Yu. V. Malykhin and K. S. Ryutin},
     title = {The {Product} of {Octahedra} is {Badly} {Approximated} in the $\ell_{2,1}${-Metric}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {85--90},
     publisher = {mathdoc},
     volume = {101},
     number = {1},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2017_101_1_a6/}
}
TY  - JOUR
AU  - Yu. V. Malykhin
AU  - K. S. Ryutin
TI  - The Product of Octahedra is Badly Approximated in the $\ell_{2,1}$-Metric
JO  - Matematičeskie zametki
PY  - 2017
SP  - 85
EP  - 90
VL  - 101
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2017_101_1_a6/
LA  - ru
ID  - MZM_2017_101_1_a6
ER  - 
%0 Journal Article
%A Yu. V. Malykhin
%A K. S. Ryutin
%T The Product of Octahedra is Badly Approximated in the $\ell_{2,1}$-Metric
%J Matematičeskie zametki
%D 2017
%P 85-90
%V 101
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2017_101_1_a6/
%G ru
%F MZM_2017_101_1_a6
Yu. V. Malykhin; K. S. Ryutin. The Product of Octahedra is Badly Approximated in the $\ell_{2,1}$-Metric. Matematičeskie zametki, Tome 101 (2017) no. 1, pp. 85-90. http://geodesic.mathdoc.fr/item/MZM_2017_101_1_a6/

[1] A. D. Ioffe, V. M. Tikhomirov, “Dvoistvennost vypuklykh funktsii i ekstremalnye zadachi”, UMN, 23:6 (144) (1968), 51–116 | MR | Zbl

[2] G. G. Lorentz, M. von Golitschek, Yu. Makovoz, Constructive Approximation. Advanced Problems, Grundlehren Math. Wiss., 304, Springer-Verlag, Berlin, 1996 | MR | Zbl

[3] E. M. Galeev, “Poperechniki po Kolmogorovu klassov periodicheskikh funktsii odnoi i neskolkikh peremennykh”, Izv. AN SSSR. Ser. matem., 54:2 (1990), 418–430 | MR | Zbl

[4] A. D. Izaak, “Poperechniki po Kolmogorovu v konechnomernykh prostranstvakh so smeshannoi normoi”, Matem. zametki, 55:1 (1994), 43–52 | MR | Zbl

[5] E. D. Gluskin, “Peresecheniya kuba s oktaedrom plokho approksimiruyutsya podprostranstvami maloi razmernosti”, Priblizhenie funktsii spetsialnymi klassami operatorov, Vologodskii gos. ped. in-t, Vologda, 1987, 35–41 | MR | Zbl

[6] E. M. Galeev, “Lineinye poperechniki klassov Geldera–Nikolskogo periodicheskikh funktsii mnogikh peremennykh”, Matem. zametki, 59:2 (1996), 189–199 | DOI | MR | Zbl

[7] A. D. Izaak, “Poperechniki klassov Geldera–Nikolskogo i konechnomernykh mnozhestv v prostranstvakh so smeshannoi normoi”, Matem. zametki, 59:3 (1996), 459–461 | DOI | MR | Zbl

[8] E. D. Gluskin, “Ekstremalnye svoistva ortogonalnykh parallelepipedov i ikh prilozheniya k geometrii banakhovykh prostranstv”, Matem. sb., 136 (178):1 (5) (1988), 85–96 | MR | Zbl

[9] B. S. Kashin, “Ob odnom izometricheskom operatore v $L^2(0,1)$”, C. R. Acad. Bulgare Sci., 38:12 (1985), 1613–1615 | MR | Zbl

[10] M. A. Lifshits, Gaussian Random Functions, Math. Appl., 322, Kluwer Acad. Publ., Dordrecht, 1995 | MR | Zbl

[11] R. Latala, K. Oleszkiewicz, “Gaussian measures of dilatations of convex symmetric sets”, Ann. Probab., 27:4 (1999), 1922–1938 | MR | Zbl

[12] T. Royen, “A simple proof of the gaussian correlation conjecture extended to some multivariate gamma distributions”, Far East J. Theor. Stat., 48:2 (2014), 139–145 | MR | Zbl

[13] G. Hargé, “A particular case of correlation inequality for the Gaussian measure”, Ann. Probab., 27:4 (1999), 1939–1951 | DOI | MR | Zbl

[14] A. Giannopoulos, “On some vector balancing problems”, Studia Math., 122:3 (1997), 225–234 | MR | Zbl