Is the following product-like space a Polish space?

Let $\mathcal{M}_1(\mathbb R)$ denote the space of Borel probability measures on $\mathbb R$. The space is a Polish space (a space which admits a complete, separable, metric) using, say the Levy-Prokhorov metric. For $\mu \in \mathcal{M}_1(\mathbb R)$, let $L^1(\mu)$ denote the Banach space of $\mu$-integrable functions (mod $\mu$-null). Again, for each $\mu$, $L^1(\mu)$ is a … Read more

Fractional Hajłasz-Besov-like similar to the Korevaar-Schoen-Sobolev spaces?

Suppose that (X,μ,d) and (Y,ν,ρ) are doubling metric measure spaces. Fix α>0 and define the space, analogously to this paper, as the collection of all measurable functions f:X→Y satisfying: (∫∞0[∫y∈Q∫x∈Qρ(f(x),f(y))pμ(B(x,t))αdμ(x)dμ(y)]pq1t1+sqdt)q<∞ Then the functions satisfying the above constraint can be seen as a non-Euclidean analogue of Hajłasz-Besov spaces, similar to the Korevaar-Shoen extensions of the Sobolev … Read more

universal 0-dimensional separable metric subspaces

Let  U:=(U δ)  be a separable metric space which is universal for all finite metric spaces, i.e. for every finite metric space X:=(X d)  there exists an isometric embedding of  X  into  U. Q:   Does there exist a 0-dimensional subset  C⊆U  in  U  such that space  (C δ|C×C)  is universal for all finite metric spaces? Similar questions … Read more

An inequality about metric spaces

I started studying this article(《L2 CURVATURE BOUNDS ON MANIFOLDS WITH BOUNDED RICCI CURVATURE》) about 3 months ago: In this article, there is a seemingly simple assertion on page 27 that troubled me for several days: “The statement (1) follows now easily by the combination of (3.39) and (3.41).” In fact, I think there are … Read more

Points of differentiability of squared distance from a point in metric spaces

Here the link to the same question I posted on MSE with no answer. Let $(X,d)$ be a complete and separable metric space and let $I:=(0, + \infty)$. I recall the definition of absolutely continuous curve in this setting: we say that $u \in AC(I;X)$ if there exists $g \in L^1(I)$, $g \ge0$ a.e. s.t. … Read more

When do Polish spaces admit complete metric making them CAT(κ)\mathrm{CAT}(\kappa)?

Question Let X be a Polish space. When are there known conditions under which X‘s topology can be metrized by a metric d such that (X,d) is a: CAT(κ) space with κ>0, CAT(0) space, CAT(κ) space with κ<0. For definitions/background see below… What I Known CAT(0) Case: This question is related to the following posts: … Read more

Covering number CkC^k-balls in C(Rn)C(\mathbb{R}^n)

Fix a positive integer n and and an non-negative integer k. The Arzela-Ascoli theorem guarantees that for a given positive integer k and a given L>0 the set BallCk,1([0,1]n)(0,L):={f∈C(Rn):‖ is compact in C([0,1]^n) for the topology of uniform convergence on compacts. Furthermore, by virtue of the inclusion Ball_{C^{k,1}([0,1]^n)}(0,L)\subseteq Ball_{C^{0,1}(\mathbb{R}^n)}(0,L) and the metric entropy estimates of … Read more

Finite approximations to the Kuratowski/Fréchet embedding

Let $(X,d)$ be a compact doubling metric space with doubling constant $C>0$. Let $\{\mathbb{X}_n\}_{n=0}^{\infty}$ be a sequences of finite subsets of $X$ with $$ \left\{B\left(x_k,\frac1{n}\right)\right\}_{k=0}^{\mathbb{X}_n} \mbox{covers } X \mbox{ and } \#\mathbb{X}_n \mbox{is the $\frac1{k}$-covering number of $X$}. $$ Fix some $x^{\star}\in X$ and consider the associated sequences of $1$-Lipschitz maps $$ K_n:\,x\mapsto \left(d(x,x_n)-d(x_n,x^{\star})\right)_{x_n\in \mathbb{X}_n}. … Read more