The Minimal Volume of Simplices Containing a Convex Body
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,...
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todo:paper_10506926_v29_n1_p717_Galicer2023-10-03T16:00:28Z The Minimal Volume of Simplices Containing a Convex Body Galicer, D. Merzbacher, M. Pinasco, D. Convex bodies Isotropic position Random simplices Simplices Volume ratio 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. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_10506926_v29_n1_p717_Galicer |
| institution |
Universidad de Buenos Aires |
| institution_str |
I-28 |
| repository_str |
R-134 |
| collection |
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) |
| topic |
Convex bodies Isotropic position Random simplices Simplices Volume ratio |
| spellingShingle |
Convex bodies Isotropic position Random simplices Simplices Volume ratio Galicer, D. Merzbacher, M. Pinasco, D. The Minimal Volume of Simplices Containing a Convex Body |
| topic_facet |
Convex bodies Isotropic position Random simplices Simplices Volume ratio |
| description |
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. |
| format |
JOUR |
| author |
Galicer, D. Merzbacher, M. Pinasco, D. |
| author_facet |
Galicer, D. Merzbacher, M. Pinasco, D. |
| author_sort |
Galicer, D. |
| title |
The Minimal Volume of Simplices Containing a Convex Body |
| title_short |
The Minimal Volume of Simplices Containing a Convex Body |
| title_full |
The Minimal Volume of Simplices Containing a Convex Body |
| title_fullStr |
The Minimal Volume of Simplices Containing a Convex Body |
| title_full_unstemmed |
The Minimal Volume of Simplices Containing a Convex Body |
| title_sort |
minimal volume of simplices containing a convex body |
| url |
http://hdl.handle.net/20.500.12110/paper_10506926_v29_n1_p717_Galicer |
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1782029749216870400 |