1
vote
0answers
16 views

When is the normal cone to a closed convex set in a locally convex set maximal monotone?

Let $X$ be a locally convex set with the following property: (P) $\forall C\subset X$ closed convex, the normal cone $N_C$ is maximal monotone as a multi-valued operator from $X$ to its topological ...
0
votes
0answers
19 views

Relation between Compactness and convex hull [closed]

Let (V,∥⋅∥) be a finite-dimensional normed space. Show that, if A⊆V is compact , then so is conv(A).
-2
votes
0answers
18 views

Relation between cpmpactness and convex hull [closed]

Let $\left ( V , \left \| \cdot \right \| \right )$ be a finite-dimensional normed space. Show that, if $ A \subseteq V$ is compact , then so is $conv(A)$.
0
votes
0answers
45 views

when does the Minkowski inequality for infinity norm become equality

I have two vectors $x, y \in \mathbb{R}^d$, it is well known as Minkowski inequality that: $|x+y|_\infty \leq |x|_\infty + |y|_\infty$, where $|x|_\infty= \underset{i=1..d}{\max} |x_i|$ with $x = ...
0
votes
1answer
39 views

The definition of convex body and the Hilbert cube

I currently have a question about the definition of convex body. The formal definition is: a convex body is a convex set which has non-empty interior. By non-empty interior, we meant for a set ...
2
votes
1answer
42 views

Compact set and its extreme points

I am reading Chapter 3: Convexity of Rudin's "Functional Analysis". Here is the problem I'm having trouble solving (number 18): Let $K$ be the smallest convex set in $\mathbb{ R}^3$ that contains ...
0
votes
0answers
21 views

Book/Papers for properties of convex/ uniformly convex Banach Spaces

I am looking for reference books and research articles which cover analysis of uniformly convex and strictly convex Banach spaces.
2
votes
1answer
46 views

Uniqueness of projection in a Banach space

Let $X$ be a Banach space, $M$ be a subspace of $X$ and $x \in X$ be any vector in $X$. Consider $\displaystyle \hat{x}_M=\arg \inf_{m\in M}\|x - m\|$. Under what conditions for $l_p$ norms $p = ...
1
vote
0answers
53 views

Minimum distance from a subspace in a Banach space

$X,Y,Z$ are disjoint sets of vectors in a Banach space M. $S_X,S_Y,S_Z$ denotes the subspace formed by the vectors in $X, Y$ and $Z$ respectively. $S_{YZ}$ and $S_{XZ}$ denote the subspaces formed by ...
1
vote
0answers
17 views

On Convex Interpolation and distances

Let $C$ denote the class of all real-valued convex functions on $[0, 1]^2$. Fix $n \geq 2$ and points $x_1, \dots, x_n$ in $[0, 1]^2$. Let $S \subset R^n$ be defined by \begin{equation*} S := ...
0
votes
0answers
12 views

extension of semilinear functional in cone.

I'm studying Nigel Kalton's work in extrapolation Banach space theory (paper: Differentials of complex interpolation processes for Kothe function spaces). My question is: Let $T$ be a cone contained ...
6
votes
0answers
198 views

Closest fixed point to a convex set

Consider the compact convex sets $Y \subset X \subset \mathbb{R}^n$, and a Lipschitz continuous function $f : X \rightarrow X$. Assume that $f$ has multiple fixed points. (From Brouwer's theorem, $f$ ...
0
votes
0answers
40 views

Fréchet normal cone

Given $x\in \Omega(\subset X)$ (X: Banach space) and $\varepsilon\geq 0$, the set of $\varepsilon-$normals to $\Omega$ at $x$ by \begin{align} \widehat N_\varepsilon(x;\Omega):=\left\{x^*\in X^*\mid ...
5
votes
1answer
90 views

Continuous linear image of closed, bounded, and convex set of a Hilbert Space is compact

Is my proof of this proposition correct ? And is this proposition well known? Proposition: Let $C$ be a closed, bounded, and convex set in a separable Hilbert space $H$. Let $L : H \to \mathbb{R}^n$ ...
2
votes
2answers
43 views

Is the distance between disjoint closed convex subsets of a Hilbert space positive? Is it attained?

Let $H$ be an infinite dimensional and separable Hilbert space. Let $A,B$ be infinite, closed and convex subsets of $H$. If $A$ and $B$ are disjoint and if at least one of them is bounded, is the ...
6
votes
3answers
381 views

Dual norm intuition

The dual of a norm $\|\cdot \|$ is defined as: $$\|z\|_* = \sup \{ z^Tx \text{ } | \text{ } \|x\| \le 1\}$$ Could anybody give me an intuition of this concept? I know the definition, I am using it ...
0
votes
0answers
19 views

Convex Inequality describing Functions inside specific area

Let us assume that we have two functions $f_1$, $f_2:[0,1] \rightarrow \mathbb{R}^{2}$, which describe each a point trajectory on the plane. Let us further assume that we parametrize those functions ...
0
votes
0answers
32 views

Proving concavity for complicated function

I have a rather complicated function, $f$, that I am trying to demonstrate is log-log-concave, i.e., $$\frac{d^2\log f}{(d\log x)^2}\leq 0.$$ The reason I think it is concave is purely heuristic. ...
3
votes
1answer
45 views

Generalization of log-convexity (log-concavity): log-log-convexity (log-log-concavity)?

$\underline{\mathrm{Background\; on\; function\; Convexity}}$ A function, $f$, is convex if: $$f( x\theta+y(1-\theta) ) \leq \theta f(x) + (1-\theta)f(y).$$ $f$ is concave if $-f$ is convex, [1]. If ...
1
vote
1answer
32 views

On a Banach space $X$, is the functional $x \mapsto \frac{1}{p}\|x\|^p$ convex?

Let $X$ be a Banach space. Let $p > 1$ and, consider the functional $X \to \mathbb{C}$ given by: $$x \mapsto \frac{1}{p}\|x\|^p$$ I would like the know if the above functional is convex. That ...
1
vote
1answer
37 views

Prove that a Hilbert space is convex of power type $2$

Let $X$ be a Banach space. For $\epsilon \in (0,2]$, define: $$\delta_X(\epsilon) = \inf_{x,y \in X}\{1 - \|\frac{1}{2}(x + y)\| : \|x\| = \|y\| = 1, \|x-y\| \ge \epsilon\}.$$ Then we say that $X$ ...
1
vote
0answers
37 views

A proof for Jensen’s inequality

I’m trying to prove a version of Jensen’s inequality, but I end up with the wrong result. I’d appreciate any help or comments. The theorem states: let $\varphi :{{R}^{k}}\to R$ be convex. Then for ...
1
vote
0answers
22 views

Show P is linear if it is convex and positive homogenious functional

it might be too simple but I couldnt show the second part L linear real space $P:L\rightarrow \Bbb R$ is called positive homogenious functional if for every $x\in L$ and $\alpha\ge 0$ , $P(\alpha ...
2
votes
0answers
48 views

Krein-Milman theorem

Let $E$ be a Riesz space with order unit $u$. With $$\left\|f\right\|_u: = \inf\{\lambda \in [0,\infty): -\lambda u\leq f\leq \lambda u\} $$ $E$ becomes a normed space. The following sets are subsets ...
0
votes
1answer
49 views

Proof of the Hahn-Banach separation theorem

In the proof of the Hahn-Banach separation theorem my notes claim the following: Let $X$ be a normed $\mathbf{R}$ vector space, $A,B\subset X$ be nonempty, disjoint, convex, $A$ compact and $B$ ...
3
votes
1answer
43 views

Local convexity of the topology of weak convergence of probability measures

Let $X$ be a Polish or standard Borel space, and $\mathcal P(X)$ be the space of all Borel probability measures on $X$ endowed with the topology of weak convergence. I am thinking of using ...
1
vote
0answers
69 views

Can complex-valued affine function be approximated?

If $K$ is a compact convex set of a locally convex Hausdorff space $V$ over $\mathbb{R}$ and $A(K)$ is the set of all continuous affine real-valued function on $K$, then the set of all restrictions to ...
0
votes
1answer
29 views

Why is it a borel set on the boundary of the unit ball of $E^n$?

Given $C$ convex body (compact convex set with non-empty interior points) in $E^n$ symmetric about the origin and containing the unit ball. Let $A(r)$ denote ,for every real $r >1$, the subset of ...
1
vote
1answer
45 views

what is the meaning of asphericity of convex set in a linear normed space?

I am trying to understand the definition of spherical to within $\epsilon$ for a convex body in a linear normed space, as given in Aryeh Dvoretzky's paper [1] (section 2, page 203): A convex set ...
0
votes
1answer
40 views

Show that $L^p$ norm is logarithmically convex as a function of $p$

Let $\Omega \subset \mathbb{R}^n$ be an open, bounded domain. Let $u : \Omega \to \mathbb{R}$ be measurable with $||u||_\infty < \infty$. For $p \in [1, \infty)$, define $$\Phi_u : p \mapsto ...
2
votes
1answer
59 views

Solving $ \inf \left\{ F[\nu] : \nu \in L^2 , \nu \geq 0, \int _0 ^1 \nu=1\right\}$

Let $\phi \in \mathcal ( [0,1]^2)$ symetric , can we find a solution to the following minimisation problem? $$ \inf \left\{ F[\nu] : \nu \in L^2 , \nu \geq 0, \int _0 ^1 \nu=1\right\}$$ with $$ ...
0
votes
1answer
27 views

Continuity of bounded and convex function on Hilbert space

I'm looking for the proof (or at least hints how to prove it) of theorem: Let $H$ be a real Hilbert space and let $f:H\to (-\infty,+\infty]$ be a convex function, bounded from above in some ...
0
votes
1answer
26 views

separate convex and concave function by affine function

Let $f:\mathbb R^n\to\mathbb R$ be a concave continuous function and $g:\mathbb R^n\to\mathbb R$ be a convex continuous function such that $f\leq g$. Then there exists an affine continuous function ...
0
votes
0answers
41 views

Hahn-Banach separation theorem

Let $V_1,V_2$ be convex subsets of a normed space $X$ with $V_1^\circ\neq\emptyset$ and $V_1^\circ\cap V_2 =\emptyset$. Then there exists $x'\in X'\setminus\{0\}$ such that $$\operatorname{Re} ...
1
vote
1answer
25 views

Infinite dimensional convex cones

Let $C$ be a convex cone in a topological real vector space $V$. Assume that we have a linear functional $\varphi: V \to \mathbb{R}$ such that $\varphi(x) \geq 0$ for all $x \in C$. Further assume ...
0
votes
1answer
58 views

uniqueness of Hahn-Banach extension for convex dual spaces

Let $X'$ be strict convex, i.e. for all $x_1',x_2'\in X'$ with $\|x_1'\|_{X'}=\|x_2'\|_{X'}=1$ the implication $$\left\|\frac{x_1'+x_2'}{2}\right\|=1\Rightarrow x_1'=x_2'$$ holds. In this case the ...
2
votes
0answers
39 views

Lower semicontinuity of a Bochner integral of a convex function

I'm looking for the following result: Let $\Omega \subset \mathbb{R}^n$ be a bounded domain and let $f$. The map $$u \mapsto \int_0^T \int_{\Omega} f(u(t))$$ is lower semicontinuous for $u \in ...
0
votes
0answers
22 views

Polar set and adjoint operator

I'm trying to understand the following statement: A bounded operator between Banach spaces $u:X\rightarrow Y$ satisfies: $$\textrm{inf}\{\;||u(B_X \cap S)||:S\subset X,\,\textrm{codim}S=k \; ...
2
votes
1answer
46 views

A question about norm for bounded linear transformations

Let $H$, $K$ be Banach spaces, and let $A: H \rightarrow K$ be a bounded linear transformation. Its norm is defined by: \begin{equation} \|A\| = sup\{\|Ah\|_K: \|h\|_H \le 1\} \end{equation} How to ...
2
votes
0answers
122 views

Convex subsets, Normed spaces, Separating hyperplane

I'm trying to understand the proof of a theorem taken from a textbook. Theorem: If $S_1,S_2$ are disjoint non-empty convex subsets of a real vector space $X$ (which may be infinite dimensional) then ...
0
votes
1answer
23 views

Increasing convex-like function in Hilbert space

I am intersted with the differential equation $$x'(t)=f(t,x(t)),\ t\in \mathbb{R}.$$ Can we find an example of a Hilbert space $H$ and a function $f:\mathbb{R}\times H \to H$ which satisfy the two ...
0
votes
1answer
25 views

Decreasing Function Projected onto Simplex

Consider $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$, defined as $f(x) := a x + b$, where $a<0$ and $b \in \mathbb{R}_{\leq 0}^{n}$. Note that $f$ is decreasing: $$ x \geq y \Longrightarrow f(x) ...
0
votes
0answers
30 views

What Projections preserve Pseudocontractiveness?

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a pseudocontraction, i.e., $$ \left\| f(x) - f(y) \right\|^2 \leq \left\| x-y\right\|^2 + \left\| f(x) - f(y) - (x-y) \right\|^2 $$ for all $x,y \in ...
0
votes
1answer
36 views

Are there necessary and sufficient conditions for Krein-Milman type conclusions?

This, the third of three self-answered questions, contains a proof of necessary and sufficient conditions for Krein-Milman type conclusions. The first question is here. The second question is here. ...
0
votes
1answer
17 views

Conditions for augmenting a collection of sets so that the new sets are small but the hull is large?

This is the second of three self-answered questions which will culminate in a proof of necessary and sufficient conditions for Krein-Milman type conclusions. The first question is here. The third ...
1
vote
0answers
67 views

When do partial subgradients give a subgradient?

I'm looking for sufficient conditions that guarantee that partial subgradients of a convex, lower-semicontinuous functional $f:X_1\times X_2\rightarrow\overline{\mathbb{R}}$ form a subgradient of $f$. ...
0
votes
1answer
37 views

Are there necessary and sufficient conditions so that every element in a partially ordered set is either the least element or in the upset of an atom?

This is the first of three self-answered questions which will culminate in a proof of necessary and sufficient conditions for Krein-Milman type conclusions. The second question is here. The third ...
1
vote
1answer
79 views

Strictly convex unit balls in $L^p$

I need to show that if $1<p<\infty$, then the unit ball is strictly convex in $L^p$, that is, $||\lambda x+(1-\lambda)y|| < 1$ whenever $||f|| = ||g||=1$ and $\lambda \in (0,1)$. I tried ...
0
votes
1answer
16 views

Establishing convexity of a function

Let $\theta \in \Theta \subset \mathbb{R}^k$. I have the following objective function $$ F(\theta):=||\max(0,f_1(\theta)),...,\max(0,f_n(\theta))||^2 $$ where $||\cdot||$ is the Euclidean Norm and ...
2
votes
0answers
137 views

Projection and Pseudocontraction on Hilbert space

Let $H$ be a real Hilbert space with inner product $\langle\cdot, \cdot \rangle: H \times H \rightarrow \mathbb{R}$, and induced norm $\left\| \cdot \right\|: H \rightarrow \mathbb{R}_{\geq 0}$. Let ...