0
votes
1answer
20 views

Redefine a discrete compact set

I need to find twice continuously differentiable functions $g_i: \mathbb{R}^2 \rightarrow \mathbb{R}$ $i=1,\ldots,I$ such that the set $\{ 0,1\}^2=\{(0,0), (1,0), (0,1), (1,1) \}$ can be written as ...
0
votes
1answer
22 views

Define a compact and convex set through inequality constraints

I need to find twice continuously differentiable functions $g_i: \mathbb{R}^2 \rightarrow \mathbb{R}$ $i=1,...,I$ such that the set $\{ 0,1\}^2=\{(0,0), (1,0), (0,1), (1,1) \}$ can be written as $\{x ...
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 ...
0
votes
0answers
31 views

Creating a $C^2$ smooth convex curve which joins circular arcs

Suppose that I have many points $p_1$,..., $p_n$, all very close to origin 0. For R very, very large, I have circular arcs of radius R, $\alpha_i$, about $p_i$. I do not speak about their ...
1
vote
1answer
10 views

Properties that guarantee quasiconvexity in $\mathbb{R}^n$

I have an open bounded connected domain $\Omega \subseteq \mathbb{R}^n$ and I would like to say that for every two $x,y \in \Omega$ there is a path $\gamma$ from $x$ to $y$ of length at most $C|x-y|$ ...
0
votes
1answer
21 views

Integral of increasing continuous function is convex

Suppose $g$ is increasing and continuous. Does it follow that $G(x) = \int_0^xg(y)dy$ is convex? Clearly $G'$ is increasing and continuous, and $G''\geq 0$ exists a.e., but I don't see how this ...
5
votes
1answer
95 views

A bounded subset in $\mathbb R^2$ which is “nowhere convex”?

Let $F : \mathbb S^1 \to \mathbb R^2$ represents a simple closed curve $C$ in $\mathbb R^2$. The Jordan curve theorem says that the curves bounds a interior domain $\Omega$ and $\partial \Omega= C$. ...
0
votes
0answers
20 views

Convex in $ \mathbb{R^n}$

Prove that: [A be a convexe part $(A\subseteq \mathbb{R^n})] \implies [\forall x_1,x_2,...x_n\in A ,\forall\alpha_1,\alpha_2,...\alpha_n\ge0 $ $with$ $ \ \alpha_1+\alpha_2+...+\alpha_n=1 ...
0
votes
1answer
21 views

Convex function almost surely differentiable.

If f: $\mathbb{R}^n \rightarrow \mathbb{R}$ is a convex function, i heard that f is almost everywhere differentiable. Is it true? I can't find a proof (n-dimentional). Thank you for any help
0
votes
2answers
56 views

Solution in general for a seemingly simple problem

Let $\mathbb{S}$ be a closed, bounded, convex set in $\mathbb{R}^N$. Let $\mathbb{x}=[x_1,\dots,x_N]$ be any arbitrary vector in $\mathbb{S}$. Then what can we comment on the problem \begin{align} ...
0
votes
1answer
44 views

How can the sum of two closed cones be not closed?

Can there be two closed cones $K_1$ and $K_2$ in $\mathbb{R}^3$ such that $K_1+K_2$ need not be closed?
0
votes
1answer
20 views

Intersection of strictly monotone, convex functions on interval with two given intersections

Given two functions $f,g:[0,1]\rightarrow[0,1]$ that are smooth, strictly convex, strictly monotone s.t. $f(1)=g(1)=0$, $g(0)=f(0)=1$ and $f(x_0)>g(x_0)$ holds for some point $x_0$, does it follow ...
1
vote
1answer
40 views

Sets defined by distance to a convex set

Let $Y \subset \mathbb R^n$ be a bounded convex set, let $R>0$, and let $$Z := \left\{z \in \mathbb R^n : d(z,Y) > \dfrac12R \right\}$$ where $$ d(z,Y) = \inf_{y\in Y}|y-z|. $$ If you like, ...
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 ...
0
votes
2answers
32 views

Convexity of support function

Let $C$ be a closed non-empty set, but not necessarily convex. The support function of $C$ is given by $$S(z) = \sup_{c \in C} \langle z,c\rangle. $$ Prove that this is a convex function. ...
3
votes
0answers
46 views

Convex Functions are Continuous

In W. Rudin's Principles of Mathematical Analysis, we read in Chapter 4 that real-valued functions defined on an open interval $(a,b)\subseteq\Bbb R$ are continuous (specifically, Exercise 23). I am ...
0
votes
0answers
21 views

Show that the Rosenbrock function is strictly convex for a specific region

So we know that the Rosenbrock function is a test function of sorts, but can anyone prove that a specific region is strictly convex? Rosenbrock eqn: $ f(x_{1},x_{2}) = 100(x_{2} - x_{1}^{2})^{2} + ...
0
votes
1answer
79 views

Can a Lipschitz continuous function be strictly convex?

Let $\varphi:\mathbb R^n\to\mathbb R$, and suppose for all $x,y\in\mathbb R^n$, $$\|\varphi(x)-\varphi(y)\|\leq L\|x-y\|$$ for Lipschitz constant $L$. Is it possible for such a function to satisfy ...
1
vote
2answers
73 views

Tangent line of a convex function

Let $V : \mathbb{R}^n \rightarrow \mathbb{R}$ be (strictly) convex and continuously differentiable on $\mathbb{R}^n$. Show that for any $y \in \mathbb{R}^n$ and any $x \in \mathbb{R}^n$ $$ V(y) \geq ...
1
vote
0answers
42 views

Properties of matrix functions

Can I say that a certain matrix function is absolutely continuous or monotonically increasing in $\lambda$(assuming that the matrix is a function of $\lambda$)? In other words, are these ...
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 $ ...
10
votes
2answers
251 views

Convexity of a complicated function

Let $\mathbb{S}$ be a 2-D convex set in the positive quadrant. Let us define \begin{align} y_L=\min_{(x,y)\in\mathbb{S} }y \\ y_R=\max_{(x,y)\in\mathbb{S} }y \end{align} For any positive number ...
0
votes
2answers
24 views

Convexity of product of two given functions

Let $f(x)$ be a convex function of $x$ in a given positive interval. Also assume $f(x)\geq 0$ everywhere in that interval. Is the function $g(x)=xf(x)$ convex in that interval?. Is it possible that ...
-1
votes
1answer
69 views

Prove or disprove the concavity of the function [closed]

Prove or disprove the concavity of $f$ over the following two domains. $$f(x_1,x_2)=10-2(x_2-x^{2}_{1})^{2}$$ defined either over $$S_1=\{(x_1,x_2) : -1\leq x_1 \leq 1, -1 \leq x_2 \leq ...
4
votes
1answer
49 views

Relation between convex functions

I formed the following conjecture and, since I can't find counterexamples, am trying to prove it. Let $f, g :[0,x_{max}]\rightarrow {\mathbb R}^{+}$ such that $f',g'>0$ $f'',g''>0$ ...
2
votes
0answers
52 views

On the convexity of a particular discontinuous function.

Let $f:D\to\mathbb{R}$ be defined as follows: $$ f(\mathbf{x})=\frac{a-(\mathbf{x}_N\cdot\mathbf{x}_0+x_{n+1})}{\sqrt{\mathbf{x}_N^T\mathbf{A}\mathbf{x}_N}}, $$ where ...
2
votes
1answer
29 views

On Convexity of product of a convex and a bounded function.

Let $f:\mathbb{R}^n\to\mathbb{R}$ be a real-valued, $n$-dimensional function defined as follows: $$ f(\mathbf{x}) = g(\mathbf{x})h(\mathbf{x}), $$ where $g:\mathbb{R}^n\to\mathbb{R}$ is convex, and ...
1
vote
1answer
41 views

showing $y\to |y|^{p}$ is convex $p\geq 1$

$y\to |y|^{p}$ is convex only for $p\geq 1$ and $y\in \mathbb{R}$. This function is nondifferentiable but we can see that the second derivative is nonnegative in each interval $(-\infty,0)$ and ...
1
vote
2answers
66 views

Verifying the convexity of some function

Convex function: We will say that $f:X\rightarrow R$ is convex function if for every $\lambda\in [0,1]$ and for every $x,y\in X$ ($X$ is convex space) $f(\lambda x+(1-\lambda)y)\leq\lambda ...
2
votes
1answer
56 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 $Y$ ...
1
vote
1answer
73 views

Relation between convex set and convex function

Let $E$ be an normed vector space and $A\subset E$ be a closed nonempty set. Define $$\phi(x)=\operatorname{dist}(x,A)=\inf_{a\in A}\|x-a\|$$ Prove that if $\phi$ is convex then $A$ is convex. ...
0
votes
1answer
34 views

when convex diverging functions are monotone when divided by $x$

here is a calculus question that someone asked me to help him wuth and I have no answer for him. any help or ideas? Given $f:(0,\infty) \rightarrow \mathbb{R}$ is a convex function, and ...
2
votes
2answers
60 views

Conjugate convex functions property

$E$ is normed vector space.Let $f\in E^*$ in a bounded linear functional from $E$ to $C$ and fix $x\in E$. We have $$\forall y\in E;\ \ \ \ f(y-x)\leq \frac{1}{2}\|y\|^2-\frac{1}{2}\|x\|^2$$ And I ...
1
vote
2answers
60 views

Prove $x^y>y^x$ by using convexity

For $y>x>e$, show that $x^y>y^x$. It is not hard to prove this inequality by using the monotonicity of $\frac{\ln t}{t}$. I am curious if this inequality can be proved by using convexity of ...
1
vote
4answers
85 views

Is the convex hull operator continuous?

Is the convex hull operator continuous? I am trying to prove that the CONVEX HULL OF a finite union of non-empty convex compact sets is compact. It is easy to prove that the union of compact sets is ...
2
votes
2answers
67 views

If $f$ is continuous, why is $f$ with the property $f\left(\frac{x+y}{2}\right)\le \frac{1}{2}f(x)+\frac{1}{2}f(y)$ is convex?

If $f$ is continuous, why is $f$ with the property $$f\left(\frac{x+y}{2}\right)\le \frac{1}{2}f(x)+\frac{1}{2}f(y)$$ ,where $0\le x,y\le 1$ is convex?
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 ...
5
votes
2answers
102 views

Are convex function from a convex, bounded and closed set in $\mathbb{R}^n$ continuous?

If I have a convex function $f:A\to \mathbb{R}$, where $A$ is a convex, bounded and closed set in $\mathbb{R}^n$, for example $A:=\{x\in\mathbb{R}^n:\|x\|\le 1\}$ the unit ball. Does this imply that ...
0
votes
1answer
51 views

Prove range of f',$\{f'(x),x\in X\}$ dense in $X^*$

Let $X$ be a Banach Space and let $f: X\rightarrow \Bbb R$ be a Fre'chet differentiable function. Suppose that $f$ is bounded from below on any bounded set and satisfies $lim_{||x||\rightarrow ...
1
vote
0answers
37 views

Prove that $\{(x, y): x\in ri (dom f), y >f(x)\}\subset ri (epi f)$

$f: R^n \to (-\infty, \infty]$ is a convex function. Prove that $ri (epi f)=\{(x, y): x\in ri (dom f), y >f(x)\}$ I have used definition of convex functions to prove that the right-side is ...
0
votes
2answers
57 views

proving that a function is convex

Suppose that $f(x)=\frac{1}{k}|x|^k$ where $k>1$ and $k<\infty$. $x$ here is in $\mathbb{R^n}. $ Is $f$ convex? I am trying to use the definition of convexity but it seems like I would need some ...
1
vote
1answer
82 views

Prove that $ri (epi f)=\{(x, y): x\in ri (dom f), y >f(x)\}$

$f: R^n \to (-\infty, \infty]$ is a convex function. Prove that $ri (epi f)=\{(x, y): x\in ri (dom f), y >f(x)\}$ I have used definition of convex functions to prove that the right-side is ...
2
votes
2answers
249 views

Show that $f(x)=||x||^p, p\ge 1$ is convex function on $\mathbb{R}^n$

Show that $f(x)=||x||^p, p\ge 1$ is convex function on $\mathbb{R}^n$. I have tried to use Holder's inequality, but I still cannot solve this problem. Could you help me with this problem? Thank you ...
4
votes
2answers
152 views

Product / GM of numbers, with fixed mean, increase as numbers get closer to mean.

I am trying to prove a statement which goes like this. Let $a_i$ and $b_i$ be positive real numbers where $i = 1,2,3,\ldots,n$; where $n$ is a positive integer greater than or equal to $2$, such ...
1
vote
3answers
46 views

Why is $ \left( \frac1n \sum_{i=1}^n x_i^{p+1} \right) \geq \left( \frac1n \sum_{i=1}^n x_i \right)^{p+1} $ true?

Let us suppose that $0 \leq p \leq 1$. All variables are assumed to be non-negative. The function $x \mapsto x^{p+1}$ is strictly convex upwards, so $$ \left( \frac1n \sum_{i=1}^n x_i^{p+1} ...
1
vote
1answer
49 views

Real analysis, convexity problem

The course is an elementary course on Rudin so we don't have much material on convexity. We have this problem concerning a convex subset, $C$, of $R^k$. a) show that the closure $cl(C)$ is convex. ...
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 ...
2
votes
1answer
54 views

Find the real vector $x$ which satisfies all this?

I got this after applying KKT conditions to an optimization problem. Let $\mathbf{h}$ be a given $N\times 1$ real vector. Let $\alpha$ be a real constant. We need to find $\lambda$ and the $N\times ...
0
votes
0answers
125 views

Problems in Convex Function.

I have read an article about convex function and I have problems in this article. In this article said that "It is well known that if $F(x, y)$ is increasing relative to $y$ and $y = h(x)$ is convex ...