3
votes
1answer
33 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 ...
0
votes
0answers
17 views

A question Kolmogrov's generalized inequality for projection onto convex sets

Kolmogrov's inequality says that, if $C$ is a convex set, and $P_C(x)$ is an operator for projecting point $x$ into the convex set $C$, if $z = P_C(x)$, then for any $y \in C$ we have $$ (z - y).(x - ...
1
vote
1answer
79 views

Is $\sqrt{x}$ concave?

I have function $f(x)= \sqrt(x)$. To check is it concave or convex i am checkin $f''(x). $ Which is $ -\frac{1}{4x^{\frac{3}{2}}} < 0$ So the $f(x)$ is concave. Is it correct ? And is is the same ...
4
votes
2answers
100 views

On the convexity of a particular form of integrals

EDIT: I made some critical corrections below. Let $\mathcal{H}\colon\mathbf{w}\cdot\mathbf{x}+c=0$ be a hypeplane in $\mathbb{R}^n$. Also, let $g\colon\mathbb{R}^n\to\mathbb{R}_+$, be a non-negative, ...
1
vote
1answer
59 views

On convex hulls and intersections of chains of compact sets

Let $V$ be a topological vector space, let $\{ C_i \}_{i \in I}$ be a set of compact subsets of $V$ which forms a chain with respect to inclusion. For now, assume the following stronger properties: ...
1
vote
1answer
51 views

convex hulls of convex sets plus a point

In a real vector space, consider S and T to be disjoint, nonempty, convex subsets of the vector space and let a point x lie outside either set. How would I prove the following: co({x} union S) is ...
0
votes
0answers
144 views

The convex function proof.

Can anyone help me prove convexity conditions? I know convexity conditions. The second derivative function is greater 0 first order convexity conditions. convex function conditions Because my ...
1
vote
1answer
128 views

Prove convexity of complicated rational function

Can anyone help me prove the convexity of this rational function? The man who proved the convexity of function used these facts. But I don't know this fact is correct or not. Here are the facts and ...
0
votes
1answer
68 views

Quasi-Convexity

Can I get the conclusion that the function of matrix $P$ and $Q$ \begin{equation} \mathrm{tr}\left( PQ\right) \end{equation} is a quasi-concave function for $P>0$, and $Q>0$? It is true for ...
3
votes
2answers
82 views

About the convexity of Ky Fan's norm

As we know, the Ky Fan norm is convex, and so is the Ky Fan k-norm. My question is, does this imply that the difference between them is a non-convex function, since it results from "difference between ...
1
vote
1answer
92 views

affine set definition equivalence

How to show the following definitions are identical for an affine space: $C = p + W$ where $W$ is a subspace $p$ is a vector in $\mathbb{R}^n$, and $\lambda a + (1-\lambda) b$ is in $C$ for any $a$ ...
0
votes
2answers
65 views

Banach space Lower semi-continuity (lsc) implying continuity

How to show the following: If the monotone real valued function $f(x)$ whose domain is a subset of $R$ is lower semi continuous on every point of the interior of the domain then it is continuous on ...
1
vote
1answer
116 views

directional derivative sublinear of a convex function sublinearity problem to show

How to show the following: If $f:\mathbb R^d \rightarrow \mathbb R$ is convex then its directional derivative is sublinear? Thank you...
0
votes
1answer
120 views

directional derivative of convex function sublinear proving that fact

How can we show that the directional derivative of a proper convex function on $\mathbb{R}^n$ is sublinear? Thank you!
0
votes
0answers
69 views

Convex conjugate of a function triple conjugate

How to show that: f: R^n->R $f^{*} = f^{***}$ where $f^*$ stands for the convex conjugate of the function. Thanks a lot!
0
votes
0answers
102 views

convex lsc function affine minorant theorem proof

How to show the following: f:R^n->R A lower semicontinuous convex function f equals the pointwise supremum of all its affine minorants. Thank you!
0
votes
1answer
149 views

Relative interior inclusion of convex sets

Is relative interior(C-D) = relative interior(C) - relative interior(D) where C and D are nonempty convex sets. If so please give the proof If thats not the case could you give a counterexample ...
3
votes
0answers
67 views

maps from a convex set to itself

Suppose $S\subset \mathbb{R}^n$ be a closed convex set under Euclidean topology (but not necessarily bounded, example a closed cone). Let $\mathcal{E}(S)=\{L:\mathbb{R}^n\rightarrow \mathbb{R}^n\text{ ...
1
vote
1answer
78 views

affine function definition

If we define the affine function as $f(\lambda x + (1-\lambda)y) = \lambda f(x) + (1-\lambda)f(y)$ for every $x,y \in R^d$ and $\lambda \in R$ How to show that it is equivalent to the definition ...
1
vote
1answer
132 views

Questions regarding internal and interior points for a convex subset of a topological vector space

Suppose that $X$ is a topological vector space, with a convex subset $A$. How do we show that if the vector $u$ is in the interior of $A$, then $u$ is an internal point of $A$ and if the interior of ...
0
votes
0answers
181 views

Convex Functions on 2 variables over an interval

It is required to show that $f(x) = x_1x_2$ is a convex function on $[a,ma]^T$ where $a\ge 0$ and $m\ge1$.To show convexity we need to show that for $\lambda \in [0,1]$: $f(\lambda x + (1-\lambda ...
11
votes
1answer
468 views

Elementary applications of Krein-Milman

Recall that the Krein-Milman theorem asserts that a compact convex set in a LCTVS is the closed convex hull of its extreme points. This has lots of applications to areas of mathematics that use ...
17
votes
1answer
521 views

Is $[0,1]^\omega$ homeomorphic to $D^\omega$?

Let $n\in \mathbb N$ and let $D^n$ be the closed $1$-ball in $\left(\mathbb R^n, \|\,\cdot\,\|_1\right)$. It is not too hard to show that $[0,1]^n \cong D^n$ in this case. This observation leads to ...
1
vote
1answer
272 views

Definition of an extreme set?

I have an issue with a definition in Rudin's Functional Analysis in the paragraph regarding the Krein-Milman Theorem. "Let $K$ be a subset of a vector space $X$. A nonempty set $S$ in $K$ is called ...