Tagged Questions

2
votes
0answers
24 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
66 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
29 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 ...
1
vote
2answers
199 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
146 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 ...
2
votes
1answer
92 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
111 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
75 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
158 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
136 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
76 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
104 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
1answer
296 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
108 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
126 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
182 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
58 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 ...