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The Journal of Geometric Analysis

, Volume 29, Issue 1, pp 717–732 | Cite as

The Minimal Volume of Simplices Containing a Convex Body

  • Daniel GalicerEmail author
  • Mariano Merzbacher
  • Damián Pinasco
Article
  • 61 Downloads

Abstract

Let \(K \subset {\mathbb {R}}^n\) be a convex body with barycenter at the origin. We show there is a simplex \(S \subset K\) having also barycenter at the origin such that \((\frac{\text {vol}(S)}{\text {vol}(K)})^{1/n} \ge \frac{c}{\sqrt{n}},\) 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 \(K \subset {\mathbb {R}}^n\) we show there is a simplex S enclosing Kwith the same barycenter such that
$$\begin{aligned} \left( \frac{\text {vol}(S)}{\text {vol}(K)}\right) ^{1/n} \le d \sqrt{n}, \end{aligned}$$
for some absolute constant \(d>0\). Up to the constant, the estimate cannot be lessened.

Keywords

Volume ratio Simplices Convex bodies Isotropic position Random simplices 

Mathematics Subject Classification

52A23 52A38 52A40 (Primary) 52A22 (Secondary) 

Notes

Acknowledgements

The authors are grateful to the anonymous referee for the clever insight regarding Problem 1.1 which gave origin to the previous section.

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Copyright information

© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  • Daniel Galicer
    • 1
    Email author
  • Mariano Merzbacher
    • 1
  • Damián Pinasco
    • 2
    • 3
  1. 1.Departamento de Matemática - IMAS-CONICET, Facultad de Cs. Exactas y Naturales Pab. IUniversidad de Buenos AiresBuenos AiresArgentina
  2. 2.Departamento de Matemáticas y EstadísticaUniversidad T. Di TellaBuenos AiresArgentina
  3. 3.CONICETBuenos AiresArgentina

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