All Questions
Tagged with fa.functional-analysis mg.metric-geometry
188
questions
4
votes
0
answers
145
views
Continuous extension preserving modulus of continuity
Let $X$ be a (non-empty) compact subset of $D(0,M):=\left\{x\in \mathbb{R}^n:\, \|x\|\leq M\right\}$, and let $f:X\rightarrow Y$ be uniformly continuous; for some metric space $Y$. Are there any ...
2
votes
1
answer
212
views
Is there a theory of partially-defined metric spaces?
Is there a theory of metric spaces in which the distance between a given pair of points need not be defined?
I'm aware that there is a theory of partial metric spaces, but these deal with a different ...
- 2,758
0
votes
0
answers
44
views
Let $A,B,C$ be centrally-symmetric convex bodies. What is this function $G(x,y) := \sup_{b \in B}\inf_{a \in A} a^T x - b^T y + \|a-b\|_C$?
Let $A$, $B$, and $C$ be centrally-symmeric convex bodies in $\mathbb R^n$. Note that any such set can such set induces a norm $\|\cdot\|_C$ on $\mathbb R^n$ defined by $\|x\|_C := \sup_{c \in C}c^\...
- 6,824
2
votes
0
answers
180
views
Metric on space of Borel-measurable functions
Let $(X,d_X),(Y,d_Y)$ be metric spaces and $X$ is locally-compact and fix a Borel probability measure $\nu$ on $X$. For any Borel-measurable $f:X\rightarrow Y$, let $\mathcal{K}(f,\delta)$ be the set ...
7
votes
1
answer
382
views
Why $(\mathrm{Lip}([0,1]^2))^*$ is finitely representable in 1-Wasserstein space over the plane?
In "Snowflake universality of Wasserstein spaces"" by Alexandr Andoni, Assaf Naor, and Ofer Neiman, they have the following notation:
For a metric space X they write $\mathcal{P}_1(X)$ ...
- 1,018
2
votes
0
answers
47
views
A question about strong slopes (nonsmooth analysis)
Context. I'm reading the manuscrip "Nonlinear Error Bounds via a Change of Function" by Dominique Azé and Jean-Noël Corvellec (J Optim Theory Appl 2016), and I'm having a hard time ...
- 6,824
3
votes
1
answer
280
views
BV spaces and fractals -- are they Sobolev? Besov?
Do the real-valued functions of bounded variation on $[0,1]$ belong to some Sobolev/Besov class?
What about a fractal, such as the Weierstrass function?
- 6,111
1
vote
0
answers
118
views
Metric transforms that preserve $\ell^1$ embeddability
Consider a function $f$ from reals to reals such that $f$, when applied to pairwise Manhattan distances between $n$ points, always results in a set of Manhattan distances.
Work by Schoenberg and ...
- 93
1
vote
1
answer
188
views
Metric / strong slope restriction of function on unit ball in $\mathbb R^m$
Diclaimer. I'm not sure this is the right venue for this question, but I'll give it a try
Definition [Strong / metric slope]. Given a complete metric space $(M,d)$ and a function $f:M \to (-\infty,+\...
- 6,824
2
votes
1
answer
168
views
Can we define geodesic in the space of compactly supported functions?
From Wikepedia, the definition of geodesic is stated as:
A curve $\gamma: I\to M$ from an interval $I$ of the reals to the metric space $M$ is a geodesic if there is a constant $v\geq 0$ such that ...
- 263
16
votes
1
answer
516
views
Balls in Hilbert space
I recently noticed an interesting fact which leads to a perhaps difficult question. If $n$ is a natural number, let $k_n$ be the smallest number $k$ such that an open ball of radius $k$ in a real ...
- 2,019
2
votes
0
answers
244
views
The contraction principle in quasi metric spaces
I am researching contractive mappings and I need the article of I. A. Bakhtin "The contraction principle in quasi metric spaces"(1989) or at least part where explanation is given for ...
12
votes
5
answers
1k
views
Examples of metric spaces with measurable midpoints
Given a (separable complete) metric space $X=(X,d)$, let us say $X$ has the measurable (resp. continuous) midpoint property if there exists a measurable (resp. continuous) mapping $m:X \times X \to X$ ...
- 6,824
2
votes
0
answers
144
views
Explicit homeomorphism between $L^p$ and Sobolev Space
From the Anderson-Kadec theorem, we know that all separable infinite-dimensional Banach spaces are homeomorphic. I'm wondering, is there an explicit such homeomorphism between $W^{p,k}(\mathbb{R}^n)$ ...
- 5,079
4
votes
0
answers
97
views
Smoothing continuous functions in metric space
Let $(X,\rho)$ be a metric space.
For any $f:X\to\mathbb{R}$, define the local Lipschitz constant of $f$ at $x$ by
$$ \Lambda_f(x) := \sup_{x'\in X\setminus\{x\}} \frac{|f(x)-f(x')|}{\rho(x,x,')}
.
$$...
- 6,111