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Questions tagged [geometric-functional-analysis]

This tag is for questions relating to "geometric functional analysis", lies at the interface of convex geometry, functional analysis and probability. It has numerous applications ranging from geometry of numbers and random matrices in pure mathematics to geometric tomography and signal processing in engineering and numerical optimization and learning theory in computer science.

Geometric functional analysis studies high dimensional linear structures. Some examples of such structures are Euclidean and Banach spaces, convex sets and linear operators in high dimensions. A central question of geometric functional analysis is: what do typical $~n$-dimensional structures look like when $~n~$ grows to infinity ? One of the main tools of geometric functional analysis is the theory of concentration of measure, which offers a geometric view on the limit theorems of probability theory. Geometric functional analysis thus bridges three areas – functional analysis, convex geometry and probability theory.

Applications: Ideas and methods of geometric functional analysis found a number of applications in computer science, especially in high-dimensional randomized algorithms. In many cases, when deterministic algorithms do not exist or are unknown, the methods relying on measure concentration provide efficient probabilistic substitutes. This is, however, a two-way street, as computer science questions have led to purely mathematical problems which are among the most important in the whole area.

References:

https://www.math.uci.edu/~rvershyn/papers/GFA-book.pdf

44 questions
12
votes
2 answers

Given a "composite" norm, what polygon describes its unit ball?

When answering this question about finding the open unit ball $\mathscr{B} := \{ x \in \mathbb{R}^2: \| x \| < 1\}$ of the "composite" norm $$ \| \cdot \|: \mathbb{R}^2 \to \mathbb{R}, \ (x,y) \mapsto a \| (x,y) \|_1 + \frac{b}{2} \| (x,y)…
ViktorStein
  • 5,024
10
votes
1 answer

About non-separable Hilbert spaces

On Reed & Simon, vol 2, chapter X, problem 4, it is asked: Let $M$ and $N$ be closed subspaces of a separable Hilbert space. If $\dim M > \dim N$, prove that $M\cap N^{\perp} \ne \{0\}$. Here, $\dim V$ is the cardinality of a Hilbert basis for…
Rodrigo Dias
  • 4,445
7
votes
2 answers

Concentration of norm of projection onto a subspace

Let $x$ be a random vector uniformly distributed on the unit sphere $\mathbb{S}^{n-1}$. Let $V$ be a linear subspace of $\mathbb{R}^n$ of dimension $k$ and let $P_V(x)$ be the orthogonal projection of $x$ onto $V$. I have seen quoted in the…
Manos
  • 26,949
5
votes
2 answers

Geometric implication of the Sobolev embedding

It is stated in section 10 of this paper that the usual Sobolev embedding $$W^{1,1}(\mathbb{R}^n) \subset L^{n/(n-1)}(\mathbb{R}^n)$$ can be interpreted in geometrical terms as an isoperimetric statement. Although the authors said that this is…
Fei Cao
  • 2,988
5
votes
1 answer

Definition of the Berkovich spectrum

I am trying to read these notes: http://www-personal.umich.edu/~takumim/Berkovich.pdf Regarding the Berkovich spectrum. In definition [2.24] it says that the spectrum is the set of bounded (non-trivial) multiplicative semi-norms. But in the explicit…
Keen-ameteur
  • 8,404
4
votes
1 answer

For discrete group $G$ and $H\leq G$. Show that $G$ also satisfies the Folner condition if $H$ satisfies it and $[G:H]<\infty$.

A finitely generated group $G=\langle S \rangle$ is said to have the Folner condition if $\forall \varepsilon>0$, there exists a finite subset $F\subset G$ such that $$\#((S\cup S^{-1})F\setminus F)<\varepsilon\# F,$$ where $\#S$ is the cardinality…
Danilo Similatan
  • 61
4
votes
1 answer

Realizing the Berkovich affine line as a union of Berkovich spectrums

I am trying to understand what is the relation of the affine Berkovich space to the Berkovich space on an appropriate polynomial ring. A more exact version of the question is as follows: Let $(K,\Vert \cdot \Vert)$ be a complete valued field. The…
Keen-ameteur
  • 8,404
3
votes
0 answers

Extreme point of the unit ball of $K(X, Y)^*$.

Let $K(X, Y)$ be the collection of all compact linear operators from $X$ to $Y$ and $K(X, Y)^*$ be the dual of $K(X, Y)$. I am interested to know the extreme points of the unit ball of $K(X, Y)^*$. In the year 1977, Fakhouri proved that, for real…
Tuh
  • 826
3
votes
0 answers

Possible reverse triangle inequality

I'm looking at the convergence (when blowing up the metric) of the spectrum of a self-adjoint operator $P$ that acts on differential forms of a 3-dimensional closed manifold M. Let $\lambda$ be a complex (not real) number. I need to prove the…
Watanabe
  • 1,081
3
votes
0 answers

Sobolev space of differential forms

I came across the following definition: The Sobolev space $W^{k,p}_1(M)$ is the space of differential forms $\alpha\in\Omega^pM$ such that $$\|\alpha\|^2_0=\int_M\alpha\wedge \star\alpha<\infty \qquad \&\qquad \|\nabla^l\alpha\|^2_0<\infty \quad…
Watanabe
  • 1,081
3
votes
0 answers

Shorter proof that the fiber of an extreme point contains an extreme point

I think I have a proof of the following result: Let $V$ be a separable real Banach space. Let $M \subset V^*$ be a nonempty convex subset of the unit ball in $V^*$ which is closed in the weak-$*$ topology. For any $\lambda \in M$, there exist an…
Sophie M
  • 928
3
votes
1 answer

A good resource on the Radon-Nikodym Property in reflexive Banach Spaces?

I'm looking for a good resource that builds the theory of the Radon-Nikodym Property. I'm not particularly interested in the measure-theoretic characterisation; I'd like the geometry of Banach Spaces version, involving strongly exposed points of…
Theo Bendit
  • 53,568
2
votes
0 answers

How to show a point is a weak* -weak continuous for the identity map on $X^*$ or on $X^{**}$?

I am trying to understand the Remark 3.2 mentioned in the paper titled as "On Weak* -Extreme Points in Banach Spaces" written by S. Dutta and T. S. S. R. K. Rao…
Tuh
  • 826
2
votes
1 answer

Give a constructive example of a limit point of the convex hull of the unit ball in infinite dimensional Banach space.

I was studying Krein-Milman Theorem in the aspect of infinite dimensional Banach spaces. We know that the closed unit ball $B_{X^*}=\{f\in X^*:\|f\|\leq 1\}$ of dual space $X^*$ of an infinite dimensional Banach space $X$ is convex and compact under…
Tuh
  • 826
2
votes
0 answers

Given a rectangle and a equilateral triangle, is it possible to construct a line that bisects the area of the triangle and rectangle?

The pancake theorem guarantees that there is a line that can bisect a rectangle and a triangle simultaneously. The Borsuk Ulam Theorem can be used to prove that theorem. Show that any line through the center of a rectangle bisects the area of…
Dea
  • 21
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