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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

  

Source: Aubrun, Guillaume - Institut Camille Jordan, Université Claude Bernard Lyon-I

 

Collections: Mathematics