SAMPLING CONVEX BODIES: A RANDOM MATRIX APPROACH GUILLAUME AUBRUN Summary: SAMPLING CONVEX BODIES: A RANDOM MATRIX APPROACH GUILLAUME AUBRUN Abstract. We prove the following result: for any # > 0, only C(#)n sample points are enough to obtain (1+#)­approximation of the inertia ellipsoid of an unconditional convex body in R n . Moreover, for any # > 1, already #n sample points give isomorphic approximation of the inertia ellipsoid. The proofs rely on an adaptation of the moments method from the Random Matrix Theory. Warning: this version di#ers from the (to be) published one (the proof of the main theorem is actually slightly simpler here). 1. Introduction and the main results Notation kept throughout the paper: The letters C, c, C # ... denote absolute positive constants, notably independent of the dimension. The value of such constants may change from line to line. Similarly, C(#) denotes a constant depending only on the parameter #. The canonical basis of R n is (e 1 , . . . , e n ), and the Euclidean norm and scalar product are denoted by | · | and #·, ·#. The operator norm of a matrix is denoted by # · #. For a real symmetric matrix A, we write #max (A) (respectively # min (A)) for the largest (respectively smallest) eigenvalue of A. A convex body is a convex compact subset of R n with non­empty interior. A convex body K is said to be unconditional if it is invariant under sign flips of the coordinates: for any # = (# 1 , . . . , #n ) # {-1, 1} n , (x 1 , . . . , xn ) # K ## (# 1 x 1 , . . . , #n xn ) # K. We reserve the letters X,Y to denote an R n ­valued random vector; X 1 , . . . , XN are i.i.d. copies of X . If EX = 0, X is said to be centered. The random vector X is said to be isotropic if it is centered Collections: Mathematics