35 research outputs found
Improved bounds in the scaled Enflo type inequality for Banach spaces
It is shown that if (X,||.||_X) is a Banach space with Rademacher type p \ge
1, then for every integer n there exists an even integer m < Cn^{2-1/p}log n (C
is an absolute constant), such that for every f:Z_m^n --> X, \Avg_{x,\e}[||f(x+
m\e/2)-f(x)}||_X^p] < C(p,X) m^p\sum_{j=1}^n\Avg_x[||f(x+e_j)-f(x)||_X^p],
where the expectation is with respect to uniformly chosen x \in Z_m^n and \e
\in \{-1,1\}^n, and C(p,X) is a constant that depends on p and the Rademacher
type constant of X. This improves a bound of m < Cn^{3-2/p} that was obtained
in [Mendel, Naor 2007]. The proof is based on an augmentation of the "smoothing
and approximation" scheme, which was implicit in [Mendel, Naor 2007]
Nearest points and delta convex functions in Banach spaces
Given a closed set in a Banach space , a point
is said to have a nearest point in if there exists such that
, where is the distance of from . We shortly
survey the problem of studying how large is the set of points in which have
nearest points in . We then discuss the topic of delta-convex functions and
how it is related to finding nearest points.Comment: To appear in Bull. Aust. Math. So
Bourgain's discretization theorem
Bourgain's discretization theorem asserts that there exists a universal
constant with the following property. Let be Banach
spaces with . Fix and set .
Assume that is a -net in the unit ball of and that
admits a bi-Lipschitz embedding into with distortion at most
. Then the entire space admits a bi-Lipschitz embedding into with
distortion at most . This mostly expository article is devoted to a
detailed presentation of a proof of Bourgain's theorem.
We also obtain an improvement of Bourgain's theorem in the important case
when for some : in this case it suffices to take
for the same conclusion to hold true. The case
of this improved discretization result has the following consequence. For
arbitrarily large there exists a family of
-point subsets of such that if we write
then any embedding of , equipped with the
Earthmover metric (a.k.a. transportation cost metric or minimumum weight
matching metric) incurs distortion at least a constant multiple of
; the previously best known lower bound for this problem was
a constant multiple of .Comment: Proof of Lemma 5.1 corrected; its statement remains unchange
Improved bounds in the metric cotype inequality for Banach spaces
It is shown that if (X, ||.||_X) is a Banach space with Rademacher cotype q
then for every integer n there exists an even integer m< n^{1+1/q}\sum_{j=1}^n \Avg_x [ ||f(x+ (m/2) e_j)-f(x)
||_X^q ] < C m^q \Avg_{\e,x} [ ||f(x+\e)-f(x) ||_X^q ]$, where the expectations
are with respect to uniformly chosen x\in Z_m^n and \e\in \{-1,0,1\}^n, and all
the implied constants may depend only on q and the Rademacher cotype q constant
of X. This improves the bound of m< n^{2+\frac{1}{q}} from [Mendel, Naor 2008].
The proof of the above inequality is based on a "smoothing and approximation"
procedure which simplifies the proof of the metric characterization of
Rademacher cotype of [Mendel, Naor 2008]. We also show that any such "smoothing
and approximation" approach to metric cotype inequalities must require m>
n^{(1/2)+(1/q)}.Comment: 27 pages, 1 figure. Fixes a slight error in the proof of Lemma 4.3 in
the arXiv v2 and the published pape
Improved bounds in the scaled Enflo type inequality for Banach Spaces
It is shown that if (X; β₯ Β· β₯X) is a Banach space with Rademacher type p β₯ 1 then for every n β N there exists an even integer m . β²n2-1/p log n such that for every f : β€ββΏ β X, Ex;" [ βf ( x + m /2 β° ) β f(x)βββ] .β²X mp Ξ£n j=1 Ex [ β₯f(x + ej) β f(x)β₯p X ] ; where the expectation is with respect to uniformly chosen x β β€ββΏ and " β {β1; 1}βΏ. This improves a bounds of m β² nββ»β/β=p that was obtained in [7]. The proof is based on an augmentation of the \smoothing and approximation" scheme, which was implicit in [7].peerReviewe