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 ...
2
votes
0answers
38 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
21 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 \; ...
0
votes
1answer
67 views

Show homeomorphism between convex hull and unit ball?

In the proof of Schauder fixed point theorem in Evans' PDE book, he uses a claim that the convex hull $K$ of $N$ points $x_1,\dots,x_N$ in a convex compact subset $A$ of a Banach space $X$ is ...
1
vote
1answer
57 views

Invariant convex subset

Let X be a banach space and $T$ a bounded operator on X with $||T||$ less than or equal to 1. If $T$ is an isometry and $r(T)$<1 show that there is a closed convex non-zero proper subset $C$ of the ...
1
vote
0answers
35 views

The equivalent definition of denting point

How i can prove that If $K$ is a subspace of Banach space $X$, $x$ is denting point of $K$,when for every $\varepsilon>0$,there is a unit vector $x^{*}\in X^{*}$ and $\delta>0$ such that ...
3
votes
1answer
69 views

Hyperplane which can separate two closed convex set

Define: $$E=:\ell^1(R)=\{x=(x_n)_n: \|x\|_1=\sum_{n=1}^{\infty}|x_n|<\infty\}$$ We have two closed convex sets $X,Y$ as subsets of normed vector space $\ell^1(R)$ with $X\cap Y=\emptyset$ and $Y$ ...
5
votes
2answers
275 views

extreme points of the unit ball of the Schatten classes?

Suppose $1<p<\infty$. What are the extreme points of the unit ball of the Schatten classes $S^p$? See below for the definition of $S^p$: http://en.wikipedia.org/wiki/Schatten_norm
1
vote
0answers
33 views

$\operatorname{sconv}(A)=\left\{ \sum_{i=1}^\infty\lambda_i\cdot a_i:a_i\in A, \lambda_i\ge0,\sum_{i=1}^\infty\lambda_i=1\right\}$ is superconvex

Let $X$ be a Banach space and $A\subset X$ a subset bounded. Denote by $\operatorname{sconv}(A)$ the superconvex hull of $A$: $$\operatorname{sconv}(A)=\left\{ \sum_{i=1}^{\infty}\lambda_i\cdot ...
2
votes
0answers
58 views

Extensions of Carathéodory's theorem

We know about the Carathéodory's theorem which is on the convex bodies of $\mathbb{R}^d$. My question is, how far we can extend it? Is it true for say, any convex object of Banach space, or for convex ...
1
vote
1answer
154 views

Is it possible for a Strongly Convex function to be unbounded below?

Let $X$ be a non-reflexive Banach space and $f:X\rightarrow\mathbb{R}$ a $C^1$ function that is Strongly Convex, i.e. $$f(u)-f(v)\geq\langle f'(v),u-v\rangle+c\|u-v\|^2$$ where $c>0$ is constant. ...
2
votes
0answers
45 views

Mid-Point Convexity Implies Convexity in Banach Spaces? [duplicate]

Possible Duplicate: Showing that $f$ is convex Let $X$ be a real Banach space and $f:X\rightarrow \mathbb{R}$ a continuous function. We say that $f$ is Mid-Point convex if for all $x,y\in ...
2
votes
2answers
277 views

fail of cantor intersection property on closed , bounded , convex sets of integrable functions

This is from my recent homework. I am asked to find a descending nested sequence of closed , bounded , nonempty convex sets $\{D_n\}$ in $L^1(\mathbb{R})$ such that the intersection is empty , where ...
2
votes
1answer
262 views

cantor intersection theorem in banach space

Here is part of the question in my HW. Let$\ \{C_n\}\subset X $ be a bounded nested decreasing sequence of closed and convex sets. I am asked to show that $\bigcap C_n \not= \emptyset$ iff X is ...
1
vote
1answer
687 views

How to prove that the closed convex hull of a compact subset of a Banach space is compact?

Can anyone help me with this problem? Prove that if $K$ is a compact subset of a Banach space $X$, then the closed convex hull of $K$ (that is, the closure of the set of all elements of the form ...
2
votes
1answer
111 views

Cancellation of addition on convex sets

I recently found a question about a property of the Minkowski sums. However the question was not properly answered (it used a projection argument which might not be true in a general Banach space). I ...
0
votes
1answer
142 views

Ball contained in a convex cone

Let $X$ be a Banach space. Let $C\subset X$ be a closed convex cone with nonempty interior. We denote by $B_X$ and $S_X$ the unit ball and the unit sphere of $X$ respectively. Let $e\in C\cap S_X$, ...
1
vote
1answer
92 views

Proving convexity of this set in $\ell^2$

This is a follow-up to the question I posted earlier this week. Consider, for a fixed sequence $(a_n)_n\in\ell^2$ the subspace $$C=\{(x_n)_{n}\in\ell^2 : |x_n|\le a_n\text{ for all ...
4
votes
1answer
277 views

Cancellation law for Minkowski sums

Let $(X,\|\cdot\|)$ be a Banach space and $A,B,C\subset X$ closed bounded non-empty convex subsets. Let $+$ denote the Minkowski symbol for addition. Does the $+$ satisfy: $$A+C\subset B+C\implies ...
2
votes
1answer
223 views

Convex function on Banach space

Let $(Y,\|\cdot\|)$ a Banach space and $b\colon Y\to \mathbb{R}$ a nonnegative convex function such that, for some $\mathcal{E}>0$, the set $\{y\in Y\,:\, b(y)<\mathcal{E}\}$ is nonempty and ...
1
vote
1answer
98 views

Where to find information on the Hilbert cube in $\ell^2$

The Hilbert cube $H$ in $\ell^2=\ell^2(\mathbb{R})$ is the subset given by $$H=\lbrace(x_n)=(x_1,x_2,\ldots)\in\ell^2:|x_n|\le2^{-n} \text{ for }n=1,2,\ldots\rbrace.$$ I've heard that ...
5
votes
2answers
168 views

Are countable intersections of convex sets convex?

Let $X$ be a Banach space $\{C_n\colon n\in\mathbb N\}$ a collection ofconvex sets in $X$. Is the set $$C=\bigcap_{n\in\mathbb N}C_n$$ convex?
1
vote
2answers
507 views

Convex hull of the extreme points of the unit ball in $C(K)$ where $K$ is the Cantor set.

Let $K$ denote the usual $1/3$ Cantor set and let $B=B_{C(K)}$ (here $B_{C(K)}$ = {$v \in C(K) : \|v\| \leq 1 $} denotes the closed unit ball of $C(K))$. Then how to prove that $B$ coincides with the ...
1
vote
1answer
128 views

Hilbert space $H$ is strictly smooth

I am trying to show that every Hilbert space $H$ is strictly smooth with modulus of smoothness $\phi_H(t)=\sqrt{1+t^2} -1 $. To show this I think I should show $H$ is uniformly smooth first. ...
2
votes
1answer
229 views

A question about the coercivity of a lsc and convex function.

I was doing a proof and I need to show a result to conclude it: $X$ is a reflexive Banach space with a norm, $\|\cdot\|$, of class $\mathcal{C}^1$. $f:X\to\overline{\mathbb{R}}$ is lower ...
3
votes
1answer
328 views

Differentiability of Moreau-Yosida approximation.

I want to show that if $X$ is a reflexive Banach space with norm of class $\mathcal{C}^1$ and $f\colon X\to\mathbb{R}\cup \{+\infty\}$ is convex and lower semicontinuous, then $f_{\lambda}$ is ...
3
votes
0answers
72 views

Using convexity and separation to prove bounds on norm bound functionals

I'm quoting here a homework problem with two clauses. I've already managed to find a solution for the first clause, and have problems generalizing it for the second clause, I'll go into details after ...