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)

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The largest volume ratio of given convex body K ⊂ R n is defined as lvr(K) := supwhere the sup runs over all the convex bodies L. We prove the following sharp lower bound c √ n ≤ lvr(K), for every body K (where c > 0 is an absolute constant). This result improves the former best known lower bound, of order n log log(n) . We also study the exact asymptotic behavior of the largest volume ratio for some natural classes. In particular, we show that lvr(K) behaves as the square root of the dimension of the ambient space in the following cases: if K is the unit ball of an unitary invariant norm in R d×d (e.g., the unit ball of the p-Schatten class S d p for any 1 ≤ p ≤ ∞), K is the the unit ball of the full/symmetric tensor product of ℓp-spaces endowed with the projective or injective norm or K is unconditional.