Sur l'intégrabilité de norme dans un espace de Banach
The One-Third-Trick and Shift Operators
We obtain a representation as martingale transform operators for the rearrangement and shift operators introduced by T. Figiel. The martingale transforms and the underlying sigma algebras are obtained explicitly by combinatorial means. The known norm estimates for those operators are a direct consequence of our representation.
The random paving property for uniformly bounded matrices
This note presents a new proof of an important result due to Bourgain and Tzafriri that provides a partial solution to the Kadison-Singer problem. The result shows that every unit-norm matrix whose entries are relatively small in comparison with its dimension can be paved by a partition of constant size. That is, the coordinates can be partitioned into a constant number of blocks so that the restriction of the matrix to each block of coordinates has norm less than one half. The original proof of...
Thin-shell concentration for convex measures
We prove that for s < 0, s-concave measures on ℝⁿ exhibit thin-shell concentration similar to the log-concave case. This leads to a Berry-Esseen type estimate for most of their one-dimensional marginal distributions. We also establish sharp reverse Hölder inequalities for s-concave measures.
Une propriété du convexe topologique vague des probabilités normales sur un espace topologique
Uniform minimality, unconditionality and interpolation in backward shift invariant subspaces
We discuss relations between uniform minimality, unconditionality and interpolation for families of reproducing kernels in backward shift invariant subspaces. This class of spaces contains as prominent examples the Paley-Wiener spaces for which it is known that uniform minimality does in general neither imply interpolation nor unconditionality. Hence, contrarily to the situation of standard Hardy spaces (and of other scales of spaces), changing the size of the space seems necessary to deduce unconditionality...
Uniformly accurate quantile bounds via the truncated moment generating function: the symmetric case.
Volumetric invariants and operators on random families of Banach spaces
The geometry of random projections of centrally symmetric convex bodies in is studied. It is shown that if for such a body K the Euclidean ball is the ellipsoid of minimal volume containing it and a random n-dimensional projection is “far” from then the (random) body B is as “rigid” as its “distance” to permits. The result holds for the full range of dimensions 1 ≤ n ≤ λN, for arbitrary λ ∈ (0,1).
Weak Distances between Random Subproportional Quotients of
Lower estimates for weak distances between finite-dimensional Banach spaces of the same dimension are investigated. It is proved that the weak distance between a random pair of n-dimensional quotients of is greater than or equal to c√(n/log³n).
Weak Grothendieck's theorem.