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Results tagged with mg.metric-geometry
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user 78539
Euclidean, hyperbolic, discrete, convex, coarse geometry, metric spaces, comparisons in Riemannian geometry, symmetric spaces.
1
vote
Mathematics of doodling and the winding number
Your formula for area (or I should say volume form), is always true up to within an error of size $o(r)$. The geometry (boundary, curvature, etc.), only comes into play when you try to write down the …
- 6,853
3
votes
Accepted
Covering number of $l_2$ Ball in $\mathbb{R}^d$
The $\epsilon$-covering number of the euclidean unit ball in $\mathbb R^d$ scales like $(1/\epsilon)^d$. More formally,
Lemma. If $\epsilon < 1$, then $(1/\epsilon)^d \le N(\epsilon, B_2) \le (3/\ …
- 6,853
7
votes
How to think about dual space of a certain space of Lipschitz functions
This would be a really long comment, so I've decided to post it as an answer. Hope it helps!
Disclaimer. I'm still learning FA, and my answer is based on my blurred understanding of the subject. I hop …
- 6,853
5
votes
1
answer
899
views
Hausdorff distance is a lower (or upper bound) for what probability metric?
In a metric space $X=(X, d)$, given a probability measure $\mu$ and two subsets $A$ and $B$ of positive measure, it's not hard to prove that
$$
d(A, B) \le W(\mu|_A, \mu|_B),
$$
where
$d(A, B):= \i …
- 6,853
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 $m:X \times X \to X$ …
- 6,853
2
votes
0
answers
48
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 understanding a cer …
- 6,853
1
vote
1
answer
417
views
Growth rate of bounded Lipschitz functions on compact finite-dimensional space
Let $\mathcal X$ be a metric space of diameter $D$ and "dimension" (e.g doubling dimension) $d$. Let $L \in [0, \infty]$ and $M \in [0, \infty)$ and consider the class $\mathcal H_{M,L}$ of $L$-Lipsch …
- 6,853
0
votes
0
answers
44
views
Let $A,B,C$ be centrally-symmetric convex bodies. What is this function $G(x,y) := \sup_{b \...
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^\to …
- 6,853
1
vote
1
answer
202
views
Conditions for Lipschitzness of boundary normal vector, almost everywhere
Let $C$ be a nonempty closed subset of $\mathbb R^n$. It is known that any such set satisfies the following condition
(Unique CPP a.e). For almost every $x \in \mathbb R^n$, there exists a unique poi …
- 6,853
1
vote
1
answer
189
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,+ …
- 6,853
2
votes
Metric / strong slope restriction of function on unit ball in $\mathbb R^m$
Disclaimer. I'm going to answer post an answer, since I'm probably the only one interested in this problem...
So, after rethinking my problem, I'm fine with replacing the unit-sphere by its convex ha …
- 6,853
2
votes
Accepted
L1 distance between gaussian measures
Explicit upper and lower bounds are obtained in Theorem 1.2 and Proposition 2.1 of The total variation distance between
high-dimensional Gaussians.
- 6,853