All Questions
Tagged with measure-concentration geometric-measure-theory
9 questions
2
votes
0
answers
202
views
Prove or disprove that $u=0$ a.e. on $\Bbb R^d$
Let $\Omega\subset\Bbb R^d$ be an open set. Let $k:\Bbb R^d\to [0,\infty)$ be measurable such that $0\in \operatorname{supp}k$. This implies that $\Omega\subset \Omega_k=\Omega+\operatorname{supp}k$. ...
- 1,101
3
votes
1
answer
190
views
Example where concentration of measure fails nontrivially
A metric probability space $(X, \mu, \rho)$, i.e., a complete separable metric space with a probability measure on its Borel sets, is said to satisfy (Gaussian) concentration of measure property if ...
- 141
1
vote
1
answer
249
views
(Novel?) notion of concentration/dispersion
Consider the measurable space $(\Omega,\mathscr{B})$ endowed with two positive measures: a "volume $\nu$" and a probability measure $\mu$. For example, one might take $\Omega=\mathbb{R}^n$ (with the ...
- 6,541
1
vote
0
answers
105
views
Gaussian isoperimetry for $\ell_p$ norms
Let $\gamma_n$ be the standard Gaussian measure on $\mathbb R^n$. It is well-known (e.g see Proposition 1) that for a given Gaussian volume content, half-spaces $H=\{x \in \mathbb R^n | a^Tx \le b\}$ ...
- 6,853
2
votes
0
answers
92
views
Lower bound to $\epsilon$-expansion of a subset of a half-sphere
Below are two known lemmas on a $d$-dimensional sphere (related to the isoperimetric inequality). I would like to know: does a similar statement like this holds for a $d$-dimensional dome also (i.e. ...
- 21
7
votes
1
answer
424
views
Transportation-cost inequality for pushforward measure
Let $X=(X,d_X)$ and $Y=(X,d_Y)$ be metric spaces and $\varphi: X\rightarrow Y$ be an $L$-Lipschitz map, with $0 \le L < \infty$. Suppose $\mu$ is a probability measure on $X$ which satisfies ...
- 6,853
7
votes
1
answer
344
views
Level sets of weakly differentiable funtions
Let $C$ be a $C^1$ hypersurface in $R^n$ and let $u \in C^1(R^n)$. Suppose
$$\nabla u(x) \cdot \eta(x)=|\nabla u| \ \ \forall x\in C$$
where $\eta(x)$ is the normal vector to $C$ at $x$ ($\nabla u$ ...
1
vote
0
answers
151
views
Does Newtonian capacity increase strictly when mass is spread?
We start with two disjoint compact sets A and B with positive capacities. Then, we translate B s.t. $B+rv$ is disjoint from A and B and ,more importantly, $dist(x,y)<dist(x,y+rv)$ for all $x\in A$ ...
- 5,474
11
votes
2
answers
3k
views
Levy's isoperimetric inequality for sphere
Let me recall subj:
If $s>0$, $A$ and $B$ are two subsets of $\mathbb{S}^{n}$, $|A|=|B|$ ($|\cdot|$ stands for the Lebesgue measure on the sphere) and $B$ is a cup $B=\{ (x_1,x_2,\dots,x_n)\in \...
- 109k