Volume ratio for the Cartesian product of convex bodies
Algebra i analiz, Tome 32 (2020) no. 5, pp. 114-129.

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A. I. Khrabrov. Volume ratio for the Cartesian product of convex bodies. Algebra i analiz, Tome 32 (2020) no. 5, pp. 114-129. http://geodesic.mathdoc.fr/item/AA_2020_32_5_a4/

[1] Bakharev F. L., “Ekstremalno dalekie normirovannye prostranstva s dopolnitelnymi ogranicheniyami”, Mat. zametki, 79:3 (2006), 339–352 | MR | Zbl

[2] Bakharev F. L., “Otsenki maksimalnykh rasstoyanii mezhdu prostranstvami, normy kotorykh invariantny otnositelno zadannykh grupp operatorov”, Zap. nauch. semin. POMI, 333, 2006, 33–42 | Zbl

[3] Bakharev F. L., “Obobschenie nekotorykh klassicheskikh rezultatov na sluchai modifitsirovannogo rasstoyaniya Banakha–Mazura”, Zap. nauch. semin. POMI, 333, 2006, 17–32 | Zbl

[4] Gluskin E. D., “Diametr kompakta Minkovskogo primerno raven $n$”, Funkts. anal. i ego pril., 15:1 (1981), 72–73 | MR | Zbl

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

[6] Gryunbaum B., Etyudy po kombinatornoi geometrii i teorii vypuklykh tel, Nauka, M., 1971

[7] Gurarii V. I., Kadets M. I., Matsaev V. I., “O rasstoyaniyakh mezhdu konechnomernymi analogami prostranstv $L^p$”, Mat. sb., 170:4 (1966), 481–489

[8] Kashin B. S., “Poperechniki nekotorykh konechnomernykh mnozhestv i klassov gladkikh funktsii”, Izv. Akad. Nauk SSSR, 41:2 (1977), 334–351 | MR | Zbl

[9] Kashin B. S., “O parallelepipedakh naimenshego ob'ema, soderzhaschikh vypukloe telo”, Mat. zametki, 45:2 (1989), 134–135 | MR | Zbl

[10] Khrabrov A. I., “Otsenki rasstoyanii mezhdu summami prostranstv $\ell^p_n$”, Vestn. S.-Peterburg. un-ta. Ser. 1, 2000, no. 3, 56–62 | MR

[11] Khrabrov A. I., “Ekstremalnye ob'emnye otnosheniya dlya summ normirovannykh prostranstv”, Probl. mat. anal., 21 (2000), 264–275 | Zbl

[12] Khrabrov A. I., “Obobschennye ob'emnye otnosheniya i rasstoyanie Banakha–Mazura”, Mat. zametki, 70:6 (2001), 918–926 | MR | Zbl

[13] Khrabrov A. I., “Rasstoyaniya mezhdu prostranstvami s bezuslovnymi bazisami”, Probl. mat. anal., 23 (2001), 206–220 | Zbl

[14] Artstein-Avidan S., Giannopoulos A., Milman V., Asymptotic geometric analysis, v. I, Math. Surveys Monogr., 202, Amer. Math. Soc., Providence, RI, 2015 | DOI | MR | Zbl

[15] Ball K., “Volumes of sections of cubes and related problems”, Geometric aspects of functional analysis (1987–88), Lecture Notes in Math., 1376, Springer, Berlin, 1989, 251–260 | DOI | MR

[16] Bárány I., Füredi Z., “Approximation of the sphere by polytopes having few vertices”, Proc. Amer. Math. Soc., 102:3 (1988), 651–659 | DOI | MR | Zbl

[17] Bezdek K., Litvak A. E., “On the vertex index of convex bodies”, Adv. Math., 215:2 (2007), 626–641 | DOI | MR | Zbl

[18] Bezdek K., Classical topics in discrete geometry, CMS Books Math., Springer-Verlag, New York, 2010 | DOI | MR | Zbl

[19] Brazitikos S., Chasapis G., Hioni L., “Random approximation and the vertex index of convex bodies”, Arch. Math. (Basel), 108:2 (2017), 209–221 | DOI | MR | Zbl

[20] Chevet S., “Séries de variables aléatoires gaussiennes à valeurs dans $E\mathbin{\widehat{\otimes}_\varepsilon\!}F$”, Application aux produits d'espaces de Wiener abstraits, 19, École Polytech., Palaiseau, 1978 | MR

[21] Galicer D., Merzbacher M., Pinasco D., Asymptotic estimates for largest volume ratio of a convex body, 2019, arXiv: 1901.00771v2 | MR

[22] Giannopoulos A., Hartzoulaki M., “On the volume ratio of two convex bodies”, Bull. London Math. Soc., 34:6 (2002), 703–707 | DOI | MR | Zbl

[23] Geiss S., “Antisymmetric tensor products of absolutely $p$-summing operators”, J. Approx. Theory, 68:3 (1992), 223–246 | DOI | MR | Zbl

[24] Gluskin E. D., Litvak A. E., “Asymmetry of convex polytopes and vertex index of symmetric convex bodies”, Discrete Comput. Geom., 40:4 (2008), 528–536 | DOI | MR | Zbl

[25] Gluskin E. D., Litvak A. E., “A remark on vertex index of the convex bodies”, Geometric aspects of functional analysis, Lecture Notes in Math., 2050, Springer, Heidelberg, 2012, 255–265 | DOI | MR | Zbl

[26] Gordon Y., “Some inequalities for Gaussian processes and applications”, Israel J. Math., 50:4 (1985), 265–289 | DOI | MR | Zbl

[27] Gordon Y., Meyer M., Pajor A., “Ratios of volumes and factorization through $\ell_\infty$”, Illinois J. Math., 40:1 (1996), 91–107 | DOI | MR | Zbl

[28] Gordon Y., Litvak A. E., Meyer M., Pajor A., “John's decomposition in the general case and applications”, J. Differential Geom., 68:1 (2004), 99–119 | DOI | MR | Zbl

[29] John F., Extremum problems with inequalities as subsidiary conditions, Intersci. Publ. Inc., New York, NY, 1948, 187–204 | MR

[30] Klartag B., “On convex perturbations with a bounded isotropic constant”, Geom. Funct. Anal., 16:6 (2006), 1274–1290 | DOI | MR | Zbl

[31] Macbeath A. M., “A compactness theorem for affine equivalence-classes of convex regions”, Canad. J. Math., 3:1 (1951), 54–61 | DOI | MR | Zbl

[32] Paouris G., “Concentration of mass on convex bodies”, Geom. Funct. Anal., 16:5 (2006), 1021–1049 | DOI | MR | Zbl

[33] Pełczyński A., Szarek S. J., “On parallelepipeds of minimal volume containing a convex symmetric body in $\mathbb{R}^n$”, Math. Proc. Cambridge Philos. Soc., 109:1 (1991), 125–148 | DOI | MR

[34] Pisier G., The volume of convex bodies and Banach space geometry, Cambridge Tracts Math., 94, Cambridge Univ. Press, Cambridge, 1989 | MR | Zbl

[35] Rogers C. A., Shephard C., “The difference body of a convex body”, Arch. Math. (Basel), 8 (1957), 220–233 | DOI | MR | Zbl

[36] Rogers C. A., Shephard C., “Convex bodies associated with a given convex body”, J. London Math. Soc., 33 (1958), 270–281 | DOI | MR | Zbl

[37] Rudelson M., “Estimates of the weak distance between finite-dimensional Banach spaces”, Israel J. Math., 89:1–3 (1995), 189–204 | DOI | MR | Zbl

[38] Stein S., “The symmetry function in a convex body”, Pacific J. Math., 6 (1956), 145–148 | DOI | MR | Zbl

[39] Szarek S. J., “On Kashin's almost Euclidean orthogonal decomposition of $\ell_1^n$”, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys., 26:8 (1978), 691–694 | MR | Zbl

[40] Tomczak-Jaegermann N., Banach–Mazur distances and finite-dimensional operator ideals, Pitman Monogr. Surveys Pure Appl. Math., 38, Longman Sci. Tech., Harlow–New York, 1989 | MR | Zbl