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.

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1answer
65 views

Stone's Lemma for disjoint convex subsets

Lemma: If A and B are disjoint convex sets in a linear space X, then there are complementary convex sets C containing A and D containing B. I am genuinely confused by this proof given in the book ...
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59 views

Factorization theorem (change of density)

I am starting to take an interest in convex geometry and stumbled on the following theorem due to Pisier, the proof should be in https://link.springer.com/content/pdf/10.1007/BF01450929.pdf, although ...
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46 views

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 ...
3
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0answers
36 views

Sobolev space of differential forms [duplicate]

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 \...
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2answers
122 views

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 ...
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37 views

What is the Lebesgue measure of this set?

Recently, I read Rolf Schneider's "Convex Bodies: The Brunn–Minkowski Theory". I was confused in the proof of theorem 4.1.1. Theorem 4.1.1 Let $(K_j)_{j∈N}$ be a sequence in $\mathcal{K}^n$ ...
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1answer
40 views

Variance of Geometric Distribution in terms of successes only [duplicate]

$$ Prove: var(X) = \frac{1-p}{p^2} $$ I solved for $$E[X^2]-E[X]^2$$ I did the following for $E(X)$ $$ E(X) = (\frac{1}{p}) $$ I did the following for $E(X^2)$ $$E(X^2)=\sum^\infty_n n^2p(1-p)^n$$ $$E(...
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1answer
35 views

Euler Lagrange Equation and Besse Conjecture

Many paper said that Einstein Hilbert functional $E(g)$ defined as follows $$E(g) = \int_{M} R_{g}dM_{g}$$ If it restricted on unit volume. The Euler Lagrange can be writen as $$Ric - \frac{R}{n} = ...
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1answer
43 views

Prove the existence of $\Psi\in(\ell_\infty(G))^\ast$ satisfying the amenability conditions

Let $G$ be a finitely generated group satisfying the Folner condition and let $S\subset G$ be a finite generating set of $G$. Denote by $\rho=\{\rho_g\}_{g\in G}$ the right translation action of $G$ ...
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1answer
58 views

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

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)...
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1answer
102 views

Extreme Points of Disc Algebra

I know that the extreme points of the closed unit ball of $\mathcal{H}^\infty(\mathbb{D})$, the space of all bounded holomorphic functions on the unit disc are the functions $f\in\mathcal{H}^\infty(\...
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1answer
142 views

What is Banach-Dieudonne theorem?

Let $X$ be a separable Banach space and $X^*$ be its dual and $w^*$ be weak$^*$ topology on $X^*$. Let $B(X^*)$ denotes closed unit ball of $X^*.$ I'm reading a research paper in which we want to ...
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0answers
48 views

Convergence to an $\ell_p$ ball, of Steiner symmetrization of compact convex subsets of $\mathbb R^n$

Context. I'm working on a problem, and it seems Steiner symmetrization might just be the golden trick. But first, I must make sure the process will converge to an $\ell_p$ ball... Fix $p \in [1,\...
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0answers
58 views

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-$*$ ...
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1answer
51 views

Sharp radius for univalent convex functions

Context Koebe 1/4 Theorem states the following: Theorem.- Let $f \in \mathcal{S}$, that is, the set of univalent (analytic and injective) with $f(0)=0$ and $f’(0)=1$ functions from $\mathbb{D}=\...
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35 views

refering something as "non-linear" when there is no underlying linear structure

Can I talk about a non-linear shape functional. I understand a shape functional $J$ as some mapping that takes a shape and returns a real (or complex) value. I would like to talk about a non-linear ...
2
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1answer
175 views

Applications of Sard's Theorem.

I am writing my Bachelors Thesis about Sard's Theorem and I was asking myself if there are any good applications of it or the direct consequences (Whitneys Embedding and Morse functions) in physics or ...
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1answer
48 views

Proving three convex surfaces intersect using 2 Dimensional intermediate value theorem

I have three convex functions $f_1(x,y), f_2(x,y),$ and $f_3(x,y)$. I know that $f_1(x,y)$ is non decreasing on both $x$ and $y$, and $f_3(x,y)$ is non increasing on both $x$ and $y$. $f_2(x,y)$ is ...
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0answers
82 views

Homeomorphism between a piece of sphere and a convex set on a Banach space.

Let $(X,|\cdot|)$ be a real Banach space and $K\subseteq X$ a closed cone, that is, $K$ is a nonempty closed subset of $X$ satisfying: (a) $0\in K$; (b) $x+y\in K$ whenever $x,y\in K$; (c) $tx\in K$ ...
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1answer
71 views

Proving that the norm of a linear and surjective operator is 1 (Step in Figiel's Theorem)

In the book Geometric Nonlinear Functional Analysis by Y.Benyamini and J.Lindenstrauss, in Theorem 14.2 (page 343), due to Figiel, the authors prove that a certain operator has norm $1$. This ...
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1answer
339 views

Given a "composite" norm, what polygon described 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) \...
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1answer
260 views

Hahn Banach Theorem implying existence of a nonzero linear functional taking 0 in a linear subspace

I am reading this paper. In the proof of theorem 1, it is stated By the Hahn-Banach theorem, there is a bounded linear functional on $C(I_n)$, call it $L$, with the property that $L\ne 0$ but $L(R)...
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1answer
60 views

Prove that $\forall t \in \mathbb R$ the set $f^{-1}(\{t\})$ is a hyperplane of $X$

Exercise : Let $X$ be a vector space and $f:X \to \mathbb R$ be a linear functional. Show that for all $t \in \mathbb R$, the set $f^{-1}(\{t\})$ is a hyperplane of $X$. Attempt : I have proved a ...
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0answers
185 views

Distance between two functions in term of a third function

I am wondering if it is possible to define a distance between two real-valued functions that express the concept of "the space between them that can be filled by a third function", where the third ...
4
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1answer
124 views

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-...
4
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1answer
158 views

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,\...
2
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0answers
107 views

Differentiability of Norms of $l_{\infty}$

In the book Fabian and others I saw exercise: "Let $\|$.$\|_{\infty}$ denote the canonical of $l_{\infty}$ and set $p(x) = \limsup |x_i|$. Define $\||x\|| = \|x\|_{\infty} + p(x)$ for $x \in l_{\...
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1answer
205 views

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 $M\cap N^{\perp} \ne \{0\}$. ...
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1answer
155 views

How to vary a second order function with respect to the metric tensor?

Can anybody help me to prove this relation, how is it is valid ? \begin{equation} \frac{\delta}{\delta g^{\mu\nu}}\nabla_{\sigma}\Bigr(\alpha(x^{\beta})\,\frac{\nabla^{\sigma}{\phi(x^\beta)}}{\phi(x^\...
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0answers
169 views

Bound degrees of sparse random graphs

I might be wrong but I think this problem (Exercise 2.4.2) means $d=o(\log n)$? If so, can anyone give a hint instead of telling the answer.
3
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1answer
176 views

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 ...
7
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2answers
511 views

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 ...