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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.


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
Página de inicio:717
Página de fin:732
Título revista:Journal of Geometric Analysis
Título revista abreviado:J Geom Anal


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---------- 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.
---------- 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.
---------- 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.
---------- VANCOUVER ----------
Galicer, D., Merzbacher, M., Pinasco, D. The Minimal Volume of Simplices Containing a Convex Body. J Geom Anal. 2019;29(1):717-732.