Artículo

Galicer, D.; Merzbacher, M.; Pinasco, D."The Minimal Volume of Simplices Containing a Convex Body" (2019) Journal of Geometric Analysis. 29(1):717-732
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Abstract:

Let K⊂ Rn be a convex body with barycenter at the origin. We show there is a simplex S⊂ K having also barycenter at the origin such that (vol(S)vol(K))1/n≥cn, where c> 0 is an absolute constant. This is achieved using stochastic geometric techniques. Precisely, if K is in isotropic position, we present a method to find centered simplices verifying the above bound that works with extremely high probability. By duality, given a convex body K⊂ Rn we show there is a simplex S enclosing Kwith the same barycenter such that(vol(S)vol(K))1/n≤dn,for some absolute constant d> 0. Up to the constant, the estimate cannot be lessened. © 2018, Mathematica Josephina, Inc.

Registro:

Documento: Artículo
Título:The Minimal Volume of Simplices Containing a Convex Body
Autor:Galicer, D.; Merzbacher, M.; Pinasco, D.
Filiación:Departamento de Matemática - IMAS-CONICET, Facultad de Cs. Exactas y Naturales Pab. I, Universidad de Buenos Aires, Buenos Aires, 1428, Argentina
Departamento de Matemáticas y Estadística, Universidad T. Di Tella, Av. Figueroa Alcorta 7350, Buenos Aires, 1428, Argentina
CONICET, Buenos Aires, Argentina
Palabras clave:Convex bodies; Isotropic position; Random simplices; Simplices; Volume ratio
Año:2019
Volumen:29
Número:1
Página de inicio:717
Página de fin:732
DOI: http://dx.doi.org/10.1007/s12220-018-0016-4
Handle:http://hdl.handle.net/20.500.12110/paper_10506926_v29_n1_p717_Galicer
Título revista:Journal of Geometric Analysis
Título revista abreviado:J Geom Anal
ISSN:10506926
CODEN:JGANE
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_10506926_v29_n1_p717_Galicer

Referencias:

  • Artstein-Avidan, S., Giannopoulos, A., Milman, V.D., (2015) Asymptotic Geometric Analysis, Part I, 202. , American Mathematical Society, Providence
  • Alonso-Gutiérrez, D., On the isotropy constant of random convex sets (2008) Proc. Am. Math. Soc., 136 (9), pp. 3293-3300
  • Ball, K., Volume ratios and a reverse isoperimetric inequality (1991) J. Lond. Math. Soc., 2 (2), pp. 351-359
  • Barthe, F., On a reverse form of the Brascamp-Lieb inequality (1998) Invent. Math., 134 (2), pp. 335-361
  • Brazitikos, S., Giannopoulos, A., Valettas, P., Vritsiou, B.-H., (2014) Geometry of Isotropic Convex Bodies, 196. , American Mathematical Society, Providence
  • Blaschke, W., Über affine geometrie iii: Eine minimumeigenschaft der ellipse. Berichte über die Verhandlungen der königl. sächs (1917) Gesellschaft Der Wissenschaften Zu Leipzig, 69, pp. 3-12
  • Bourgain, J., On the distribution of polynomials on high dimensional convex sets (1991) Geometric Aspects of Functional Analysis, pp. 127-137. , Milman LA, (ed), Springer, New York
  • Brazitikos, S., Brascamp-Lieb inequality and quantitative versions of Helly’s theorem (2017) Mathematika, 63 (1), pp. 272-291
  • Chakerian, G., Minimum area of circumscribed polygons (1973) Elem. Math., 28, pp. 108-111
  • Dvoretzky, A., Rogers, C.A., Absolute and unconditional convergence in normed linear spaces (1950) Proc. Natl. Acad. Sci., 36 (3), pp. 192-197
  • Gardner, R.J., (1995) Geometric Tomography, 1. , Cambridge University Press, Cambridge
  • Giannopoulos, A., Hartzoulaki, M., On the volume ratio of two convex bodies (2002) Bull. Lond. Math. Soc., 34 (6), pp. 703-707
  • Giannopoulos, A., Perissinaki, I., Tsolomitis, A., John’s theorem for an arbitrary pair of convex bodies (2001) Geom. Dedicata, 84 (1-3), pp. 63-79
  • Gross, W., Über affine geometrie xiii: Eine minimumeigenschaft der ellipse und des ellipsoids (1918) Ber. Verh. Sächs. Akad. Wiss. Leipz. Math.-Nat. Wiss. Kl, 70, pp. 38-54
  • Gruber, P., (2007) Convex and Discrete Geometry, 336. , Springer, Berlin
  • Hudelson, M., Klee, V., Larman, D., Largest j-simplices in d-cubes: some relatives of the Hadamard maximum determinant problem (1996) Linear Algebra Appl., 241, pp. 519-598
  • Kanazawa, A., On the minimal volume of simplices enclosing a convex body (2014) Arch. Math., 102 (5), pp. 489-492
  • Klartag, B., Kozma, G., On the hyperplane conjecture for random convex sets (2009) Israel J. Math., 170 (1), pp. 253-268
  • Klartag, B., On convex perturbations with a bounded isotropic constant (2006) Geom. Funct. Anal. GAFA, 16 (6), pp. 1274-1290
  • Kannan, R., Lovász, L., Simonovits, M., Isoperimetric problems for convex bodies and a localization lemma (1995) Discret. Comput. Geom., 13 (3-4), pp. 541-559
  • Kuperberg, W., On minimum area quadrilaterals and triangles circumscribed about convex plane regions (1983) Elem. Math., 38, pp. 57-61
  • Lassak, M., On the Banach-Mazur distance between convex bodies (1992) J. Geom., 41, pp. 11-12
  • Lassak, M., Approximation of convex bodies by centrally symmetric bodies (1998) Geom. Dedicata, 72 (1), pp. 63-68
  • Macbeath, A.M., An extremal property of the hypersphere (1951) Mathematical Proceedings of the Cambridge Philosophical Society, 47, pp. 245-247. , Cambridge University Press, Cambridge
  • Jiří, M., (2002) Lectures on Discrete Geometry, 108. , Springer, New York
  • McKinney, J.A.M.E.S.R., On maximal simplices inscribed in a central convex set (1974) Mathematika, 21 (1), pp. 38-44
  • Milman, V.D., Pajor, A., Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space (1989) Geometric Aspects of Functional Analysis, pp. 64-104. , In:., Springer, New York
  • Naszódi, M., Proof of a conjecture of Bárány (2016) Katchalski and Pach. Discret. Comput. Geom., 55 (1), pp. 243-248
  • Pach, J., Agarwal, P.K., (2011) Combinatorial Geometry, 37. , Wiley, New York
  • Pelczynski, A., Structural theory of Banach spaces and its interplay with analysis and probability (1983) Proceedings of the ICM, pp. 237-269
  • Pivovarov, P., On determinants and the volume of random polytopes in isotropic convex bodies (2010) Geom. Dedicata, 149 (1), pp. 45-58
  • Paouris, G., Pivovarov, P., Random ball-polyhedra and inequalities for intrinsic volumes (2017) Monatshefte für Mathematik, 182 (3), pp. 709-729
  • Pelczynski, A., Szarek, S., On parallelepipeds of minimal volume containing a convex symmetric body in R n (1991) Mathematical Proceedings of the Cambridge Philosophical Society, 109, pp. 125-148. , Cambridge University Press, Cambridge
  • Sas, E., Über eine extremumeigenschaft der ellipsen (1939) Compos. Math., 6, pp. 468-470
  • Schneider, R., Weil, W., (2008) Stochastic and Integral Geometry, , Springer, Berlin

Citas:

---------- APA ----------
Galicer, D., Merzbacher, M. & Pinasco, D. (2019) . The Minimal Volume of Simplices Containing a Convex Body. Journal of Geometric Analysis, 29(1), 717-732.
http://dx.doi.org/10.1007/s12220-018-0016-4
---------- CHICAGO ----------
Galicer, D., Merzbacher, M., Pinasco, D. "The Minimal Volume of Simplices Containing a Convex Body" . Journal of Geometric Analysis 29, no. 1 (2019) : 717-732.
http://dx.doi.org/10.1007/s12220-018-0016-4
---------- MLA ----------
Galicer, D., Merzbacher, M., Pinasco, D. "The Minimal Volume of Simplices Containing a Convex Body" . Journal of Geometric Analysis, vol. 29, no. 1, 2019, pp. 717-732.
http://dx.doi.org/10.1007/s12220-018-0016-4
---------- VANCOUVER ----------
Galicer, D., Merzbacher, M., Pinasco, D. The Minimal Volume of Simplices Containing a Convex Body. J Geom Anal. 2019;29(1):717-732.
http://dx.doi.org/10.1007/s12220-018-0016-4