0
votes
0answers
21 views

Convex cones and positively homogenous and subadditive functional

Let $V$ be a linear topological space, $K \subsetneq V$ a convex, closed cone with $0 \in K$ and $k \in K \setminus (-K)$. Show that the functional $\varphi : V \to \mathbb R \cup \{ \pm \infty \}$ ...
2
votes
2answers
39 views

Prove: $f: (a,b) \rightarrow \mathbb{R}$ is convex iff for every triple $x_1,x,x_2 \in (a,b)$…

Assignment: Prove that a function $f : (a,b) \rightarrow \mathbb{R}$ is convex if and only if for every triple $x_1,x,x_2 \in (a,b)$ with $x_1 < x < x_2$ following inequality is satisfied: ...
2
votes
1answer
44 views

Let $f: [a,b] \rightarrow R$ be convex. If f is not constant, then the supremum of $f$ is not inside of [a,b].

I could use some help with this proof. Let $a,b \in \mathbb{R}, \ a<b \ \ f: [a,b] \rightarrow \mathbb{R}$ be convex. Prove: If f is not constant, then the point, where the supremum of ...
1
vote
1answer
20 views

Tangent cone of graph and epigraph sets.

Let us first recall the definition of tangent cone $\; T(\bar x; \Omega)$ of a subset $\Omega$ at $\bar x \in \Omega$, where $X$ is a Banach space: $$T(\bar x; \Omega)=\{v\in X:\; \; \exists ...
1
vote
1answer
29 views

Convex Sets and extreme supports

Let the set $S$ in $R^n$ consists of the origin $0$ and $n$ lineary independent vectors $T_1, \ldots, T_n$. Show that $C(S)$, the convex hull of of $S$, is the intersection of its extreme supports, ...
1
vote
1answer
35 views

The closednees in Moreau - Rockafellar Theorem.

One says that $x\mapsto \langle x^*,x\rangle +\alpha\;$ is an affine minorant of $f: \; X\to \overline{\mathbb R}\;$ if $\;\langle x^*,x\rangle +\alpha \leq f(x)\;$ for all $x\in X$. The Moreau - ...
0
votes
1answer
13 views

Volume of Minkowski sum of a point and a hypercube

Let $A$ be a single point and $B$ a unit cube in $\mathbb{R}^n$, what is then the volume $\lambda \mapsto \mathrm{Vol}\big((1-\lambda)A + \lambda B\big)$? I am not exactly sure, what the set ...
0
votes
1answer
36 views

Volume of unit n-dimensional ball, definite integal

As a part of an assignment to calculate the volume of unit n-dimensional ball I got to the following expression, which I believe is true: ...
0
votes
1answer
14 views

Continuity of convex functions at point out of domain

I have been studying the continuity of a convex function and having a trouble below: In some books, the authors defined the the continuity of a convex function $f$ even $f$ is not in $\mbox{dom }f$, ...
2
votes
1answer
30 views

Distance of convex combination of pairs of points in $\mathbb{R}^n$

Given 4 points $w,x,y,z \in \mathbb{R}^n$ define for $t\in [0,1]$ $f(t)=d(wt + (1-t)x, yt + (1-t)z)$. Is this function convex? I have found a proof by differentiating twice and calculating a lot but ...
0
votes
0answers
21 views

Lipschitz in R^1 implies Lipschitz along any line in R^k (convex function)

I'm trying to solve a homework problem (baby rudin course) and realized that Lipschitz continuity really makes the problem easier for me. However, since it is not defined in Rudin I have to prove the ...
1
vote
1answer
60 views

Level set of convex functions

Let $f:\mathbb R^n \to\mathbb R \cup\{+\infty\}$ be a proper convex function, assume that there exists $c\in\mathbb R$ such that the $c$-level set $L_{\leq c}=\{x\in R^n: f(x)\leq c\}$ is nonempty and ...
5
votes
1answer
107 views

Inequality related with concave property

Assume that $f>0,f'<0$ and $f$ is logconcave(the log of $f$ is concave) and twice differentiable. Can we prove, or give a counter example to the following claim: there exists $\bar x>0$ such ...
1
vote
1answer
45 views

Caratheodorys lemma proof

I have to proof caratheodorys lemma for my oral exam. The proof is given here. I dont get the last part. "This process can be repeated until x is represented as a convex combination of at most d + 1 ...
0
votes
1answer
36 views

Concave function of two variables restricted to one variable

Suppose $u(x,y)$ is a concave and strictly increasing $\mathcal{C}^2$ function (think of a utility function from economics). Define the one variable function $f(x)=u(x,e^r(K-x))$ for all $x\in ...
1
vote
1answer
83 views

Finding the dual cone

Looking at the example here, I'm trying to understand how the author finds the dual cone $K^*$. The question asks to find the dual cone of $\{Ax | x \succeq 0\}$ where $A \in \mathbb R^{m\ ...
1
vote
1answer
97 views

Proving that a Hessian Matrix is positive definite

I'm currently stuck on a problem for my Artificial Intelligence class. The assignment is provided at the following link: http://courses.engr.illinois.edu/cs440/HW1.pdf The problems that I'm stuck on ...
2
votes
0answers
35 views

Conjugate of a function

Yes this is homework. Given $f(x) = 1^{T}(x)_+$ where $(x)_+ = \max\{0,x\}$, what is $f^*$? I know that the conjugate of a function $f$ is $f^*(y) = \sup (y^Tx - f(x))$ but I do not know how to show ...
2
votes
1answer
73 views

Determining if a function is convex

Yes this is homework. For which values of $a$ is the function $f(x)=e^{-a \sqrt x}$ with $\mathbf dom f = \mathbf R_+$ convex? The possible answers are: $a \le 0$ $a \ge 0$ $-1 \le a \le 1 $ ...
3
votes
1answer
28 views

Calculate Dq(x)

Let A be a symmetric $m \times m$ matrix, and $q(x)=x\cdot Ax$ a quadratic form on $\mathbb{R}^m$. Question: Calculate $Dq(x)$; write your answer in vector notation. Does anyone knows the answer on ...
2
votes
0answers
41 views

Find the Polar of a set.

Let $A \subset \mathbb{R}^{n}$ be non-empty. The set $$ A^{\circ} = \lbrace c \in \mathbb{R}^{n} |\; \langle c,x \rangle \leq 1 \quad \forall x \in A \rbrace $$ is called the polar of $A$. I'm ...
1
vote
1answer
55 views

Normed space problem

I am currently dealing with the following problem: Imagine you have two points $x,y$ in a normed space $(X,||.||)$ and in a convex set $K \subset X$. Now you know that $B_{\varepsilon}(x) \subset ...
1
vote
0answers
52 views

Introduction to Analysis: Convexity

A friend and I were trying to figure out this problem from our assignment. Prove that on an open $I$, a geometrically convex function $f(x)$ is continuous. To better assist the audience, it is ...
0
votes
1answer
69 views

Confusion solving constant function

Find $f:\mathbb{R}\to\mathbb{R}$ which is not a constant function which is neither star-concave, nor star-convex, but both concave and convex. Please help me how to solve for this function?
0
votes
2answers
33 views

Proving that a certain set is convex

I'm trying to prove that the set $W = \{x \in \mathbb{R}^2 : 2x_{1}^{2} + 3x_{2}^{2} \leq 4\}$ is convex. I've been trying to do this using the definition of W being convex when for all $x, y \in W $ ...
2
votes
1answer
41 views

Proof inequality using convexity

I struggling with proofing an inequality. We have to show that $x - y \le (1-\theta)^{-1} x^\theta (x^{1-\theta} - y^{1-\theta})$ holds for all $x, y > 0, \theta \in [0, 1)$. Further we know that ...
0
votes
1answer
46 views

Proof that a ball is convex when $p =\infty$

I want to prove that a ball for infinity norm is convex: $$ B_\infty=\{x\in\mathbb R^n : \|x\|_\infty\le1\} $$ I came up with this proof and appreciate it if someone can help to verify if this is ...
1
vote
1answer
83 views

Prove that $f$ is a convex function if $f=d(x,C)$ and $C$ is convex.

Question: Suppose $V$ is a normed space and $f:V \rightarrow \mathbb R$ defined as $f(x)=d(x,C)=\inf_{c \in C}||x-c||_V$ where $C$ is a convex subset of $V$. Prove $f$ is a convex function. Attempt ...
0
votes
1answer
74 views

homework about convex set

Let $C$ be a nonempty convex subset of $\mathbb R^k$. Let $x\in\mathbb R^k$. Assume that $x$ is not an interior point of $C$. Show that there exists a vector $a$ not equal to $0$ such that a'x ...
0
votes
1answer
51 views

Measure of a set

Let $A \subset R$ such that for all open interval I, $m^* (A \cap I) < 1/2 L(I)$, where L is the length of a interval and $m^*$ is measure, prove that $m^*(A)=0$. I appreciate any hint to solve ...
3
votes
1answer
74 views

Lovasz Extension Intuition

I am confused by the definition of Lovasz extension. The problem is I don't get the intuition behind the definition. In addition, Lovasz extension can be defined in different ways I don't see that ...
1
vote
2answers
50 views

How to show that $ Ax \le b$ is convex?

For $$ A \in \mathbb{R}^{m \times n}, b \in \mathbb{R}^m, c \in \mathbb{R} $$ one has to show that $$ K:= \{ x \in \mathbb{R}^n: Ax \le b \}$$ is convex. Now I'm aware that by definition, a set ...
2
votes
1answer
562 views

Is $f(x,y) = x^2y + x y^2$ (quasi-) concave or convex?

I should analyze whether the function $$f(x,y) = x^2y + x y^2 \text{ where } x,y > 0$$ is (quasi-) concave or convex. Thus, as usual, I set up the Hessian as $$ D^2f(x,y) = \left( ...
1
vote
1answer
104 views

Determining the convex hull of the union of two polyhedra

I'm doing an introductory course to linear programming and I'm working through some exercises to prepare for the final exam, I'm stuck on an exercise and I would really appreciate a hint: Let ...
0
votes
1answer
38 views

Example on Correspondences

Giva an example of correspondence $F : \mathbb{R} \rightarrow \mathbb{R}$ such that the closure of $F$ is $ \overline{F}: \mathbb{R} \rightarrow \mathbb{R}$, upper semi continuous on $\mathbb{R}$, ...
2
votes
0answers
56 views

Linear Difference Equations

Let $g_1, g_2 \in \mathbb{R}^{[0,5]}$ be defined by $g_1(x) = x$ and $g_2(x) = 1-x$ for each $x \in [0,5]$. Find a second-order homogeneous linear difference equation on $[0,3]$ such that {$g_1, g_2 ...
2
votes
1answer
57 views

How to prove that is a cone

I have to prove that the set $$K=\lbrace u\in C[0,1]\mid u(t)\geq a(t)||u|| \text{ on } [0,1]\rbrace,$$ where $a(t)=\displaystyle\frac{t(2p-t)}{p^2}$ with $\frac12<p<1$, and ...
2
votes
1answer
152 views

Prove the concavity of $F(x,y) =\ln(x)+y$ by arguing the definition of concavity.

I need help with this problem. Prove the concavity of $F(x,y) = \ln(x) + y$ by arguing the definition of concavity. A function $f$ is concave is for any $x_0, x_1 \in \mathbb{R}^2$ and $t \in ...
0
votes
1answer
61 views

Projection: two closed convex sets

I am really struggling with this problem: $C$ and $D$ are closed, convex subsets of ${R}^n$ with non-empty intersection, i.e. $C \cap D \neq \emptyset $ . Is it true that projection $p_{C\cap ...
3
votes
1answer
488 views

Why is this composition of concave and convex functions concave?

Please forgive my ignorance. I have a quick silly question about a statement given without proof in Convex Optimization by Boyd and Vandenberghe (page 87). Suppose $\mathbb{R}_+^n$ is the set of ...
1
vote
1answer
55 views

Is the set of all concave functions a convex set?

How can I prove this? I saw a similar question here: (But this was only for when g(x) is ≥0) Prove that a set defined by concave functions on $R^n$ is convex
1
vote
1answer
93 views

proving compactness and convexity of a set

Suppose functions $f(x)$ and $g(x)$ are continuous with domain $X \subset \mathbb{R} $ which is nonempty, convex and compact, can we show that $$S \equiv (f(x), g(x)) $$ for all $x \in X$ is ...
0
votes
0answers
47 views

Is $C(B^c)$ an open set?

Assume that $B$ is an open set, if $C:=\{x=\sum_{i=1}^{n}\lambda_{i}x_{i},\lambda_{i} \geq 0,\sum_{i=1}^{n}\lambda_{i}=1,x_i \in B\}$ is a convex that contains $B$, $C$ is an open set? What's ...
2
votes
0answers
56 views

Prove that $\text{int}(\text{dom}(f))$ is a convex set.

Let $f$ be a convex function. I have to prove that $\text{int}(\text{dom}(f))$ is a convex set. (Be careful with $-∞$ )
2
votes
2answers
265 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
240 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
578 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 ...
3
votes
2answers
557 views

Projection onto closed convex set

Show that the function defined by $f(t)=|P_{D}(x+td)-x|$ is nondecreasing, where $D$ is closed convex, $x\in D$, $t\geq 0$, $d\in \mathbb{R}^{n}$ and $P_{D}$ is projection onto D. I tried to solve ...
1
vote
3answers
174 views

How to show convexity of a ball in metric space?

If $(X,\|\cdot\|)$ is a normed linear space, then how to show any ball $B(x,r)$ is convex? I know that if $x,y\in A\subset V$ then $[x,y]\subset A$, where $A$ is a convex subset of vector space $V$ ...
3
votes
3answers
692 views

Using the definition of a concave function prove that $f(x)=4-x^2$ is concave (do not use derivative).

Let $D=[-2,2]$ and $f:D\rightarrow \mathbb{R}$ be $f(x)=4-x^2$. Sketch this function.Using the definition of a concave function prove that it is concave (do not use derivative). Attempt: ...