D. Azagra, R. Fry, L. Keener
Let $X$ be a separable real Hilbert space. We show that for every Lipschitz function $f:X\rightarrow\mathbb{R}$, and for every $ε>0$, there exists a Lipschitz, real analytic function $g:X\rightarrow\mathbb{R}$ such that $|f(x)-g(x)|\leq ε$ and $\textrm{Lip}(g)\leq \textrm{Lip}(f)+ε$.
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Let $X$ be a separable Banach space with a separating polynomial. We show that there exists $C\geq 1$ (depending only on $X$) such that for every Lipschitz function $f:X\rightarrow\mathbb{R}$, and every $ε>0$, there exists a Lipschitz, real analytic function $g:X\rightarrow\mathbb{R}$ such that $|f(x)-g(x)|\leq ε$ and $\textrm{Lip}(g)\leq C\textrm{Lip}(f)$. This result is new even in the case when $X$ is a Hilbert space. Furthermore, in the Hilbertian case we also show that $C$ can be assumed to be any number greater than 1.
Suppose that S is a left amenable semitopological semigroup. We prove that if ${T_{t}: t \in S}$ is a uniformly k-Lipschitzian semigroup on a bounded closed and convex subset C of a Hilbert space and $k<\sqrt{2}$, then the set of fixed points of this semigroup is a Hölder continuous retract of C. This gives a qualitative complement to the Ishihara-Takahashi fixed point existence theorem.
Twisted Hilbert spaces defined by bi-Lipschitz maps
本研究通过证明从$[0,\infty)$到$\mathbb{R}$的双Lipschitz映射子集可锥化，得到与其对偶同构但不与共轭对偶同构的奇异扭Hilbert空间$Z(\varphi)$的无限维锥，并刻画了Kalton-Peck空间，部分解答相关猜想。
Negative type and bi-lipschitz embeddings into Hilbert space
本文提出失真 p-负型概念，建立其与到 Hilbert 空间的双 Lipschitz 嵌入等价的理论，并通过二部图和 Hamming 立方体给出实例。
Lipschitz and Hölder Continuity in Reproducing Kernel Hilbert Spaces
本文系统研究了再生核希尔伯特空间中Lipschitz与Hölder连续性的充分条件及相应再生核的刻画，并整理了已有结果，为该主题提供了统一参考。
Let $B_E$ be the open unit ball of a complex finite or infinite dimensional Hilbert space. If $f$ belongs to the space $\mathcal{B}(B_E)$ of Bloch functions on $B_E$, we prove that the dilation map given by $x \mapsto (1-\|x\|^2) \mathcal{R} f(x)$ for $x \in B_E$, where $\mathcal{R} f$ denotes the radial derivative of $f$, is Lipschitz continuous with respect to the pseudohyperbolic distance $ρ_E$ in $B_E$, which extends to the finite and infinite dimensional setting the result given for the classical Bloch space $\mathcal{B}$. In order to provide this result, we will need to prove that $ρ_E(z…
On some new metric characterisations of Hilbert spaces
本文综述了用 Lipschitz 映射替代线性有界映射后，巴拿赫空间刻画为希尔伯特空间的同构版本新结果，重点关注由内积诱导等价范数的存在性。
Order-Preserving Extensions of Hadamard Space-Valued Lipschitz Maps
本文研究了在Hadamard空间中保持顺序的Lipschitz映射扩展问题，发现当维度大于等于2时，只有在特定条件下才能保持单调性，从而推翻了Kirszbraun定理的顺序理论扩展可能性。
We prove global and local versions of the so-called div-curl-lemma, a crucial result in the homogenization theory of partial differential equations, for mixed boundary conditions on bounded weak Lipschitz domains in 3D with weak Lipschitz interfaces. We will generalize our results using an abstract Hilbert space setting, which shows corresponding results to hold in arbitrary dimensions as well as for various differential operators. The crucial tools and the core of our arguments are Hilbert complexes and related compact embeddings.
Absolutely minimising Lipschitz extensions between Hilbert spaces
本文证明实希尔伯特空间闭子集到实希尔伯特空间的Lipschitz映射存在绝对极小Lipschitz延拓，回答了Naor与Sheffield的问题，还针对高维目标空间的连通域证明了密度定理，刻画了相关有界线性映射。
We analyse the relationship between different extremal notions in Lipschitz free spaces (strongly exposed, exposed, preserved extreme and extreme points). We prove in particular that every preserved extreme point of the unit ball is also a denting point. We also show in some particular cases that every extreme point is a molecule, and that a molecule is extreme whenever the two points, say $x$ and $y$, which define it satisfy that the metric segment $[x, y]$ only contains $x$ and $y$. The most notable among them is the case when the free space admits an isometric predual with some additional p…
We prove that Burenkov's Extension Operator preserves Sobolev spaces built on general Morrey spaces, including classical Morrey spaces. The analysis concerns bounded and unbounded open sets with Lipschitz boundaries in the n-dimensional Euclidean space.
We characterize preserved extreme points of Lipschitz-free spaces $\mathcal{F}(X)$ in terms of simple geometric conditions on the underlying metric space $(X,d)$. Namely, each preserved extreme point corresponds to a pair of points $p,q$ in $X$ such that the triangle inequality $d(p,q)\leq d(p,r)+d(q,r)$ is uniformly strict for $r$ away from $p,q$. For compact $X$, this condition reduces to the triangle inequality being strict. This result gives an affirmative answer to a conjecture of N. Weaver that compact spaces are concave if and only if they have no triple of metrically aligned points.
We study how well a quasi-Banach space can be coarsely embedded into a Hilbert space. Given any quasi-Banach space X which coarsely embeds into a Hilbert space, we compute its Hilbert space compression exponent. We also show that the Hilbert space compression exponent of X is equal to the supremum of the amounts of snowflakings of X which admit a bi-Lipschitz embedding into a Hilbert space.
In this paper first we describe all (not necessarily linear or bijective) transformations on $\mathbb{R}^d$ with $2\leq d<\infty$ which preserve the area of parallelograms spanned by any two vectors. We also characterize those (not necessarily linear) bijections on an arbitrary real Hilbert space that preserve the latter quantity. This answers a question raised by Rassias and Wagner, and it can be considered as a variant of the famous Wigner theorem on real Hilbert spaces which plays an important role in quantum mechanics. As a consequence, we solve a preserver problem of Molnár and Timmermann…
In this paper, we study the boundedness of the Hilbert transformation in Lorentz function spaces, thereby complementing classical results of Boyd. We also characterize the optimal range of a triangular truncation operator in Schatten-Lorentz ideals. These results further entail sharp commutator estimates and applications to operator Lipschitz functions in Schatten-Lorentz ideals.
The notion of multipliers in Hilbert space was introduced by Schatten in 1960 using orthonormal sequences and was generalized by Balazs in 2007 using Bessel sequences. This was extended to Banach spaces by Rahimi and Balazs in 2010 using p-Bessel sequences. In this paper, we further extend this by considering Lipschitz functions. On the way we define frames for metric spaces which extends the notion of frames and Bessel sequences for Banach spaces. We show that when the symbol sequence converges to zero, the multiplier is a Lipschitz compact operator. We study how the variation of parameters i…
Pełczyński's property (V$^*$) in Lipschitz-free spaces
本文证明了利普希茨自由空间中佩尔切尼茨属性(V$^*$)的局部决定性，并获得若干充分条件。证明当完全度量空间M局部紧致且纯1-不可测，或为希尔伯特空间，或属于满足双霍尔德条件的卡诺-卡拉泰多里空间类时，F(M)具有属性(V$^*$)。
Structure-Preserving Reconstruction of Convex Lipschitz Functionals on Hilbert Spaces from Finite Samples
研究在Hilbert空间上如何通过有限样本重建凸Lipschitz函数，提出了一种保持凸性和Lipschitz正则性的显式公式，并引入了凸神经函数（CNFs）作为可训练架构。
We prove that if a metric space $M$ has the finite CEP then $\mathcal F(M)\widehat{\otimes}_π X$ has the Daugavet property for every non-zero Banach space $X$. This applies, for instance, if $M$ is a Banach space whose dual is isometrically an $L_1(μ)$ space. If $M$ has the CEP then $L(\mathcal F(M),X)=\Lip(M,X)$ has the Daugavet property for every non-zero Banach space $X$, showing that this is the case when $M$ is an injective Banach space or a convex subset of a Hilbert space.
We characterize those bounded multilinear operators that factor through a Hilbert space in terms of its behavior in finite sequences. This extends a result, essentially due to S. Kwapień, from the linear to the multilinear setting. We prove that Hilbert-Schmidt and Lipschitz $2$-summing multilinear operators naturally factor through a Hilbert space. It is also proved that the class $Γ$ of all multilinear operators that factor through a Hilbert space is a maximal multi-ideal; moreover, we give an explicit formulation of a finitely generated tensor norm $γ$ which is in duality with $Γ$.
Lipschitz-free spaces over separable Banach spaces and their complementability in the bidual
研究可分Banach空间上的Lipschitz-free空间在二次对偶中的可补性，证明其可补当且仅当原空间同构于Hilbert空间，否定了Godefroy等人的相关问题，还刻画了范数1投影的存在性并推广到凸子集。
The Cahill-Casazza-Daubechies problem on Hölder stable phase retrieval
该研究探讨了无限维希尔伯特空间中相位恢复的稳定性问题，通过构造非线性子集并分析其是否满足利普希茨稳定性，揭示了霍尔德稳定性与利普希茨稳定性之间的关系。
In the setting of $\R^d$ with an $n-$dimensional measure $μ,$ we give several characterizations of Lipschitz spaces in terms of mean oscillations involving $μ.$ We also show that Lipschitz spaces are preserved by those Calderon-Zygmund operators $T$ associated to the measure $μ$ for which T(1) is the Lipschitz class $0.$
Lipschitz Estimates and an application to trace formulae
本文通过初等证明给出酉算子对的双算子积分形式的Lipschitz估计，扩展压缩算子对的算子Lipschitz估计函数类，并建立适用于特定压缩算子对的修正Krein迹公式。
Given two metric spaces $M$ and $N$ we study, motivated by a question of N. Weaver, conditions under which an isometric composition operator $C_ϕ:\mathrm{Lip}_0(M)\longrightarrow \mathrm{Lip}_0(N)$ is isometric depending on the properties of $ϕ$. We obtain a complete characterisation of those operators $C_ϕ$ in terms of a property of the function $ϕ$ in the case that $B_{\mathcal F(M)}$ is the closed convex hull of its preserved extreme points. Also, we obtain necessary and sufficient conditions for $C_ϕ$ being isometric in the case that $M$ is geodesic.
We give a simple proof of the Baillon-Haddad theorem for convex functions defined on open and convex subsets of Hilbert spaces. We also state some generalizations and limitations. In particular, we discuss equivalent characterizations of the Lipschitz continuity of the derivative of convex functions on open and convex subsets of Banach spaces.
We explicitly describe all Hilbert function spaces that are interpolation spaces with respect to a given couple of Sobolev inner product spaces considered over $\mathbb{R}^{n}$ or a half-space in $\mathbb{R}^{n}$ or a bounded Euclidean domain with Lipschitz boundary. We prove that these interpolation spaces form a subclass of isotropic Hörmander spaces. They are parametrized with a radial function parameter which is OR-varying at $+\infty$ and satisfies some additional conditions. We give explicit examples of intermediate but not interpolation spaces.
In this paper, we consider bivariate Cheney-Sharma operators which are not the tensor product construction. Precisely, we show that these operators preserve Lipschitz condition of a given Lipschitz continuous function f and also the properties of the modulus of continuity function when f is a modulus of continuity function.
Shape Derivatives of the Eigenvalues of the De Rham Complex for Lipschitz Deformations and Variable Coefficients: Part I
该研究针对变三维域上de Rham复形的特征值问题，在弱正则性假设下提供Hadamard型形状导数公式，还证明了适用于自伴算子族单/多重特征值的Helmann-Feynman定理。
In 2010, Olson \& Robinson [Transactions of the American Mathematical Society, 362(1), 145-168] introduced the notion of an almost homogeneous metric space and showed that if $X$ is a subset of a Hilbert space such that $X-X$ is almost homogeneous, then $X$ admits almost bi--Lipschitz embeddings into Euclidean spaces. In this paper, we extend this result and we show that if $X$ is a subset of a Banach space such that $X-X$ is almost homogeneous at the origin, then $X$ can be embedded in a Euclidean space in an almost bi--Lipschitz way.
We compute uniform rates of metastability for the Ishikawa iteration of a Lipschitz pseudo-contractive self-mapping of a compact convex subset of a Hilbert space. This extraction is an instance of the proof mining program that aims to apply tools from mathematical logic in order to extract the hidden quantitative content of mathematical proofs. We prove our main result by applying methods developed by Kohlenbach, the first author and Nicolae for obtaining quantitative versions of strong convergence results for generalized Fejér monotone sequences in compact subsets of metric spaces.
In this paper first we give a partial answer to a question of L. Molnár and W. Timmermann. Namely, we will describe those linear (not necessarily bijective) transformations on the set of self-adjoint matrices which preserve a unitarily invariant norm of the commutator. After that we will characterize those (not necessarily linear or bijective) maps on the set of self-adjoint rank-one projections acting on a two-dimensional complex Hilbert space which leave the latter quantity invariant. Finally, this result will be applied in order to obtain a description of such bijective preservers on the un…
A function between two metric spaces is said to be totally bounded regular if it preserves totally bounded sets. These functions need not be continuous in general. Hence the purpose of this article is to study such functions vis-á-vis continuous functions and functions that are stronger than the continuous functions such as Cauchy continuous functions, some Lipschitz-type functions etc. We also present some analysis on strongly uniformly continuous functions which were first introduced in \cite{[BL2]} and study when these functions are stable under reciprocation.
A characterization of the existence of zeros for operators with Lipschitzian derivative and closed range
本文研究了实 Hilbert 空间中具有 Lipschitzian 导数和闭值域的 $C^1$ 算子零点存在性问题，通过凸集和凸函数的条件给出了其充要条件的刻画。
Characterization of Hilbertizable spaces via convex functions
本文利用具有 Lipschitz 导数的强凸函数，或函数与其凸共轭均为 $C^2$ 的条件，给出 Banach 空间同构于 Hilbert 空间的凸分析刻画。
We prove the Sobolev-to-Lipschitz property for metric measure spaces satisfying the quasi curvature dimension condition recently introduced in E. Milman, The Quasi Curvature-Dimension Condition with applications to sub-Riemannian manifolds, Comm. Pure Appl. Math. (to appear, arXiv:1908.01513v5). We provide several applications to properties of the corresponding heat semigroup. In particular, under the additional assumption of infinitesimal Hilbertianity, we show the Varadhan short-time asymptotics for the heat semigroup with respect to the distance, and prove the irreducibility of the heat sem…
We consider the standard quantum logic ${\mathcal L}(H)$ associated to a complex Hilbert space $H$, i.e. the lattice of closed subspaces of $H$ together with the orthogonal complementation. The orthogonality and compatibility relations are defined for any logic. In the standard quantum logic, they have a simple interpretation in terms of operator theory. For example, two closed subspaces (propositions in the logic ${\mathcal L}(H)$) are compatible if and only if the projections on these subspaces commute. We present both classical and more resent results on transformations of ${\mathcal L}(H)$…
In this paper we study a new class of transformations on the set of all Hilbert space effects. This consists of the bijective maps which preserve the order and zero product in both directions. The main result of the paper gives a complete description of the structure of those transformations. As applications we obtain additional new results and some former ones as easy corollaries. In particular, we obtain the form of the ortho-order automorphisms as well as that of the sequential automorphisms. In the last paragraph of the paper we show that these two kinds of automorphisms belong to our clas…
In this note we prove that reconstruction from magnitudes of frame coefficients (the so called "phase retrieval problem") can be performed using Lipschitz continuous maps. Specifically we show that when the nonlinear analysis map $α:{\mathcal H}\rightarrow\mathbb{R}^m$ is injective, with $(α(x))_k=|<x,f_k>|^2$, where $\{f_1,\ldots,f_m\}$ is a frame for the Hilbert space ${\mathcal H}$, then there exists a left inverse map $ω:\mathbb{R}^m\rightarrow {\mathcal H}$ that is Lipschitz continuous. Additionally we obtain the Lipschitz constant of this inverse map in terms of the lower Lipschitz const…
Let $A$ be a $C^*$-algebra. Let $E$ and $F$ be Hilbert $A$-modules with $E$ being full. Suppose that $θ: E\to F$ is a linear map preserving orthogonality, i.e., $<θ(x), θ(y) > = 0$ whenever $<x, y > = 0$. We show in this article that if, in addition, $A$ has real rank zero, and $θ$ is an $A$-module map (not assumed to be bounded), then there exists a central positive multiplier $u\in M(A)$ such that $<θ(x), θ(y) > = u < x, y>$ ($x,y\in E$). In the case when $A$ is a standard $C^*$-algebra, or when $A$ is a $W^*$-algebra containing no finite type II direct summand, we also obtain the same concl…
Let $X(n),$ for $n\in\mathbb{N},$ be the set of all subsets of a metric space $(X,d)$ of cardinality at most $n.$ The set $X(n)$ equipped with the Hausdorff metric is called a finite subset space. In this paper we are concerned with the existence of Lipschitz retractions $r: X(n)\to X(n-1)$ for $n\ge2.$ It is known that such retractions do not exist if $X$ is the one-dimensional sphere. On the other hand L. Kovalev has recently established their existence in case $X$ is a Hilbert space and he also posed a question as to whether or not such Lipschitz retractions exist for $X$ being a Hadamard s…
In this paper we prove that for an arbitrary pair $\{T_1,T_0\}$ of contractions on Hilbert space with trace class difference, there exists a function $\boldsymbolξ$ in $L^1({\Bbb T})$ (called a spectral shift function for the pair $\{T_1,T_0\}$ ) such that the trace formula $\operatorname{trace}(f(T_1)-f(T_0))=\int_{\Bbb T} f'(ζ)\boldsymbolξ(ζ)\,dζ$) holds for an arbitrary operator Lipschitz function $f$ analytic in the unit disk.
Let $X$ be a Banach space and let $(ξ_j)_{j\ge 1}$ be an i.i.d. sequence of symmetric random variables with finite moments of all orders. We prove that the following assertions are equivalent: (1). There exists a constant $K$ such that $$ \Bigl(\E\Big\|\sum_{j=1}^n ξ_j f(x_j)\Big\|^2\Bigr)^{\frac12} \leq K \n f\n_{\rm Lip} \Bigl(\E\Big\|\sum_{j=1}^n ξ_j x_j\Big\|^2\Bigr)^{\frac12} $$ for all Lipschitz functions $f:X\to X$ satisfying $f(0)=0$ and all finite sequences $x_1,...,x_n$ in $X$. (2). $X$ is isomorphic to a Hilbert space.
Typical Lipschitz images of rectifiable metric spaces
本文研究可求长度量空间在典型 1-Lipschitz 映射下的像，依据切范数刻画 Hausdorff 测度保持性，并给出强可求长情形的近最优结果及范数空间中的反例。
We establish the sharp rate of continuity of extensions of $\mathbb{R}^m$-valued $1$-Lipschitz maps from a subset $A$ of $\mathbb{R}^n$ to a $1$-Lipschitz maps on $\mathbb{R}^n$. We consider several cases when there exists a $1$-Lipschitz extension with preserved uniform distance to a given $1$-Lipschitz map. We prove that if $m>1$ then a given map is $1$-Lipschitz and affine if and only if such distance preserving extension exists for any $1$-Lipschitz map defined on any subset of $\mathbb{R}^n$. This shows a striking difference from the case $m=1$, where any $1$-Lipschitz function has such p…
We study a new notion of trace operators and trace spaces for abstract Hilbert complexes. We introduce trace spaces as quotient spaces/annihilators. We characterize the kernels and images of the related trace operators and discuss duality relationships between trace spaces. We elaborate that many properties of the classical boundary traces associated with the Euclidean de Rham complex on bounded Lipschitz domains are rooted in the general structure of Hilbert complexes. We arrive at abstract trace Hilbert complexes that can be formulated using quotient spaces/annihilators. We show that, if a H…
Johnson and Lindenstrauss proved that any Lipschitz mapping from an $n$-point subset of a metric space into Hilbert space can be extended to the whole space, while increasing the Lipschitz constant by a factor of $O(\sqrt{\log n})$. We present a simplification of their argument that avoids dimension reduction and the Kirszbraun theorem.
On the equivalence of a Hessian-free inequality and Lipschitz continuous Hessian
本文证明了在广义设定下，无 Hessian 的 Jensen 型不等式与 Fréchet 可微及 Lipschitz 连续导数的等价性，其逆命题基于 Baillon-Haddad 定理得证。
We construct a finitely generated group which is an extension of two finitely generated groups coarsely embeddable into Hilbert space but which itself does not coarsely embed into Hilbert space. Our construction also provides a new infinite monster group: the first example of a finitely generated group that does not coarsely embed into Hilbert space and yet does not contain a weakly embedded expander.
Spectral and numerical radius preservers on pairs of unitarily similar operators
本文刻画了在酉相似算子对上保持谱半径或数值半径的双射线性映射，以及将酉相似变换为相似性的映射，结果覆盖有限维和无限维Hilbert空间。
Suppose that f is a Lipschitz function on the real numbers with Lipschitz constant smaller or equal to 1. Let A be a bounded self-adjoint operator on a Hilbert space H. Let 1<p<infinity and suppose that x in B(H) is an operator such that the commutator [A, x] is contained in the Schatten class S_p. It is proved by the last two authors, that then also [f(A), x] is contained in S_p and there exists a constant C_p independent of x and f such that || [f(A), x] ||_p <= C_p || [A,x] ||_p. The main result of this paper is to give a sharp estimate for C_p in terms of p. Namely, we show that C_p ~ p^2/…
This paper is concerned with the differential sensitivity analysis of variational inequalities in Banach spaces whose solution operators satisfy a generalized Lipschitz condition. We prove a sufficient criterion for the directional differentiability of the solution map that turns out to be also necessary for elliptic variational inequalities in Hilbert spaces (even in the presence of asymmetric bilinear forms, nonlinear operators and nonconvex functionals). In contrast to classical results, our method of proof does not rely on Attouch's theorem on the characterization of Mosco convergence but …
The category of Hilbert spaces and contractions has filtered colimits, and tensoring preserves them. We also discuss (problems with) bounded maps.
In this paper we prove two results regarding reconstruction from magnitudes of frame coefficients (the so called "phase retrieval problem"). First we show that phase retrievability as an algebraic property implies that nonlinear maps are bi-Lipschitz with respect to appropriate metrics on the quotient space. Second we prove that reconstruction can be performed using Lipschitz continuous maps. Specifically we show that when nonlinear analysis maps $α,β:\hat{H}\rightarrow R^m$ are injective, with $α(x)=(|\langle x,f_k\rangle |)_{k=1}^m$ and $β(x)=(|\langle x,f_k \rangle|^2)_{k=1}^m$, where $\{f_…
Density of subalgebras of Lipschitz functions in metric Sobolev spaces and applications to Wasserstein Sobolev spaces
本文证明了度量Sobolev空间中Lipschitz函数子代数在能量意义下稠密的一般准则，并将其应用于Wasserstein Sobolev空间，证明了其Hilbert性质并给出了Wasserstein梯度的显式特征与微积分法则。
We prove the following new characterization of $C^p$ (Lipschitz) smoothness in Banach spaces. An infinite-dimensional Banach space $X$ has a $C^p$ smooth (Lipschitz) bump function if and only if it has another $C^p$ smooth (Lipschitz) bump function $f$ such that $f'(x)\neq 0$ for every point $x$ in the interior of the support of $f$ (that is, $f$ does not satisfy Rolle's theorem). Moreover, the support of this bump can be assumed to be a smooth starlike body. As a by-product of the proof of this result we also obtain other useful characterizations of $C^p$ smoothness related to the existence o…
Kirszbraun extensions preserving uniform distance in Hilbert spaces
研究希尔伯特空间中基尔施布劳恩扩张保持一致距离的条件，证明相关等价条件，给出有限维目标应用及无限维或特定维度关系下的一些结论，如数据处理特征、提升定理和不等式等。
In this note we study the problem of evaluating the trace of $f(T)-f(R)$, where $T$ and $R$ are contractions on Hilbert space with trace class difference, i.e., $T-R\in\boldsymbol{S}_1$ and $f$ is a function analytic in the unit disk ${\Bbb D}$. It is well known that if $f$ is an operator Lipschitz function analytic in ${\Bbb D}$, then $f(T)-f(R)\in\boldsymbol{S}_1$. The main result of the note says that there exists a function $\boldsymbolξ$ (a spectral shift function) on the unit circle ${\Bbb T}$ of class $L^1({\Bbb T})$ such that the following trace formula holds: $\operatorname{trace}(f(T…
A Wasserstein spaces is a metric space of sufficiently concentrated probability measures over a general metric space. The main goal of this paper is to estimate the largeness of Wasserstein spaces, in a sense to be precised. In a first part, we generalize the Hausdorff dimension by defining a family of bi-Lipschitz invariants, called critical parameters, that measure largeness for infinite-dimensional metric spaces. Basic properties of these invariants are given, and they are estimated for a naturel set of spaces generalizing the usual Hilbert cube. In a second part, we estimate the value of t…
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