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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 Rn. 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 differs 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 Rn
is
(e1, . . . , en), 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 Rn
with non-empty 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
,

  

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

 

Collections: Mathematics