# Tagged Questions

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 ...
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 - ...
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 ...
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, ...
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: ...
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 ...
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 ...
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 ...
1answer
68 views

### Quasi-Convexity

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

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 ...
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 ...
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 ...