The Minimal Volume of Simplices Containing a Convex Body

Abstract

Let K ⊂ R n 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)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 very high probability.As a consequence, we provide correct asymptotic estimates on an old problem in convex geometry. Namely, we show that the simplex S min (K) of minimal volume enclosing a given convex body K ⊂ R n , fulfills the following inequality vol(S min (K)) vol(K)

MINVO Basis: Finding Simplexes with Minimum Volume Enclosing Polynomial Curves

“…Some works have focused on the properties of the smallest 𝑛simplex that encloses a given generic convex body. [Galicer et al 2019;Kanazawa 2014] derived some bounds for the volume of this simplex, while Klee showed in [Klee 1986] that any locally optimal simplex (with respect to its volume) that encloses a convex body must be a centroidal simplex. In other words, the centroid of its facets must belong to the convex body.…”

Asymptotic estimates for the largest volume ratio of a convex body

“…In [15,Theorem 1.3] the authors of this article showed that any convex body L ⊂ R n can be inscribed in a simplex S such that |S|…”

“…In other words, if S is a simplex then vr(S, L) ≪ √ n, for every convex body L ⊂ R n . Since the regular simplex is the minimal volume simplex that contains the Euclidean unit ball (see [15,Example 2.7]), by computing volumes we have √ n ≪ vr(S, B n 2 ). Therefore, for a simplex S, we know the exact asymptotic behaviour of its largest volume ratio:…”

Asymptotic estimates for the largest volume ratio of a convex body

“…In [8,Theorem 1.3] the authors of this article showed that any convex body L ⊂ R n can be inscribed in a simplex S such that |S|…”

“…In other words, if S is a simplex then vr(S, L) ≪ √ n, for every convex body L ⊂ R n . Since the regular simplex is the minimal volume simplex that contains the Euclidean unit ball (see [8,Example 2.7]), by computing volumes we have √ n ≪ vr(S, B n 2 ). Therefore, for a simplex S, we know the exact asymptotic behaviour of its largest volume ratio:…”