Markov type constants, flat tori and Wasserstein spaces

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Markov type constants, flat tori and Wasserstein spaces. / Zolotov, Vladimir.

In: Geometriae Dedicata, Vol. 195, No. 1, 01.08.2018, p. 249-263.

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Zolotov, Vladimir. / Markov type constants, flat tori and Wasserstein spaces. In: Geometriae Dedicata. 2018 ; Vol. 195, No. 1. pp. 249-263.

BibTeX

@article{f0515b17ca9b44f6b81fcfc3876320b7,

title = "Markov type constants, flat tori and Wasserstein spaces",

abstract = "Let Mp(X, T) denote the Markov type p constant at time T of a metric space X, where p≥ 1. We show that Mp(Y, T) ≤ Mp(X, T) in each of the following cases: (a) X and Y are geodesic spaces and Y is covered by X via a finite-sheeted locally isometric covering, (b) Y is the quotient of X by a finite group of isometries, (c) Y is the Lp-Wasserstein space over X. As an application of (a) we show that all compact flat manifolds have Markov type 2 with constant 1. In particular the circle with its intrinsic metric has Markov type 2 with constant 1. This answers the question raised by S.-I. Ohta and M. Pichot. Parts (b) and (c) imply new upper bounds for Markov type constants of the Lp-Wasserstein space over Rd. These bounds were conjectured by A. Andoni, A. Naor and O. Neiman. They imply certain restrictions on bi-Lipschitz embeddability of snowflakes into such Wasserstein spaces.",

keywords = "Alexandrov space, Flat manifold, Markov type, Wasserstein space",

author = "Vladimir Zolotov",

year = "2018",

month = aug,

day = "1",

doi = "10.1007/s10711-017-0287-0",

language = "English",

volume = "195",

pages = "249--263",

journal = "Geometriae Dedicata",

issn = "0046-5755",

publisher = "Springer Nature",

number = "1",

}

RIS

TY - JOUR

T1 - Markov type constants, flat tori and Wasserstein spaces

AU - Zolotov, Vladimir

PY - 2018/8/1

Y1 - 2018/8/1

N2 - Let Mp(X, T) denote the Markov type p constant at time T of a metric space X, where p≥ 1. We show that Mp(Y, T) ≤ Mp(X, T) in each of the following cases: (a) X and Y are geodesic spaces and Y is covered by X via a finite-sheeted locally isometric covering, (b) Y is the quotient of X by a finite group of isometries, (c) Y is the Lp-Wasserstein space over X. As an application of (a) we show that all compact flat manifolds have Markov type 2 with constant 1. In particular the circle with its intrinsic metric has Markov type 2 with constant 1. This answers the question raised by S.-I. Ohta and M. Pichot. Parts (b) and (c) imply new upper bounds for Markov type constants of the Lp-Wasserstein space over Rd. These bounds were conjectured by A. Andoni, A. Naor and O. Neiman. They imply certain restrictions on bi-Lipschitz embeddability of snowflakes into such Wasserstein spaces.

AB - Let Mp(X, T) denote the Markov type p constant at time T of a metric space X, where p≥ 1. We show that Mp(Y, T) ≤ Mp(X, T) in each of the following cases: (a) X and Y are geodesic spaces and Y is covered by X via a finite-sheeted locally isometric covering, (b) Y is the quotient of X by a finite group of isometries, (c) Y is the Lp-Wasserstein space over X. As an application of (a) we show that all compact flat manifolds have Markov type 2 with constant 1. In particular the circle with its intrinsic metric has Markov type 2 with constant 1. This answers the question raised by S.-I. Ohta and M. Pichot. Parts (b) and (c) imply new upper bounds for Markov type constants of the Lp-Wasserstein space over Rd. These bounds were conjectured by A. Andoni, A. Naor and O. Neiman. They imply certain restrictions on bi-Lipschitz embeddability of snowflakes into such Wasserstein spaces.

KW - Alexandrov space

KW - Flat manifold

KW - Markov type

KW - Wasserstein space

UR - http://www.scopus.com/inward/record.url?scp=85031415973&partnerID=8YFLogxK

U2 - 10.1007/s10711-017-0287-0

DO - 10.1007/s10711-017-0287-0

M3 - Article

AN - SCOPUS:85031415973

VL - 195

SP - 249

EP - 263

JO - Geometriae Dedicata

JF - Geometriae Dedicata

SN - 0046-5755

IS - 1

ER -

ID: 49951858