Convex analysis is the study of properties of convex sets and convex functions. For questions about optimization of convex functions over convex sets, please use the (convex-optimization) tag.

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Convex Hull of cyclic Permutations

It is known that the convex hull of permutation matrices yields exactly the stochastic matrices. I am interested in the convex hull of cyclic permutation matrices. Trivially this is a subset of the ...
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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 ...
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110 views

convex/concave problem.

I want to show that if $y = f(x) > 0$ is a concave function on $\mathbb{R}$, then $z = \frac{1}{f(x)}$ is a convex function. Since $f(x) > 0$ then if we applied the second derivative test, ...
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There is a closed hyperplane.

$\textbf{Question: }$ If $M$ is an open convex set in normed linear space $R$ and $x_{0}\not\in M$, then there exists a closed hyperplane which passes through the point $x_{0}$ and does not intersect ...
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Voronoi-Diagram - split concave polygon

I have a concave polygon c and a set of points p[]. What I need is to have a set of polygons splitted by the voronoi-diagram. My problem is the following: I already found a library to calculate the ...
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65 views

The convex conjugate of a quadratic form with positive semi definite matrix

I want to find the convex conjugate (Legendre transform) of a quadratic for $1/2x^{t}Qx$ when $Q$ is positive semi-definite. If Q is non-singular, the solution is easy - the gradient is $y-Qx=0$ so ...
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57 views

Cyclic monotonicity of sub-differential domain and convex property

I am looking for hints/proof's overview/reference about this proposition : Let $S\subset \mathbb{R}^d\times\mathbb{R}^d$. There exist a convex function $\phi$ such that $S\subset \partial\phi$ ...
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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 ...
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Is this set convex ?2

Is this set convex for every arbitrary $\alpha\in \mathbb R$? $$\Big\{(x_1,x_2)\in \mathbb R^2_{++} \,\Big|\, x_1x_2\geq \alpha\Big\}$$ Where $\mathbb R^2_{++}=[0,+\infty)\times [0,+\infty)$.
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strictly convex function when plotted but second derivative not unambiguously positve

I have a function $$ z(x) = (Kx)^{x/(1-x)}, x \in (0,1)\text{ and }K>1 $$ When I plot the function it has a U-shape. However when I take the second derivative wrt $x$ I have the following ...
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34 views

Space where the separation theorem doesn't hold

I have read the proof of the next separation theorem: Let X b a normed space in R and A a convex and open set that contains 0. Let b be a point wich is not in A then there exists f in X* such that ...
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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 ...
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102 views

Dual of a rational convex polyhedral cone

A rational convex polyehedral cone $\sigma\subseteq\mathbb R^n$ is a set of the form $$ \sigma=\operatorname{Cone}(u_1,\dots,u_k) := \left\{ \sum_{i=1}^k r_i u_i \,\Bigg|\, r_i\ge ...
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48 views

Convex Function to Given (Three) Data Points

Assume that a function $h(x)$ is decreasing and convex given interval $[l,u]$. I'd like to get a function which connects three points, say $(a,h(a)), (b,h(b)), (c,h(c))$, where $l\leq a<b<c\leq ...
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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 ...
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52 views

Linear combination of convex set is convex

A set $U$ is a convex set if whenever $\mathbf{x},\mathbf{y}$ are points in $U$, the line segment joining $\mathbf{x}$ to $\mathbf{y}$,$$\mathcal l(\mathbf{x},\mathbf{y})=\left\{t\mathbf ...
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infimum of a convex function over an open domain

Let $f: \cal D_0 \to [0, \infty]$ be a convex function on a compact set $\cal{D}_0$ and let $\cal D \subseteq \cal D_0$. I think the following holds: $$ \inf_{x \in \cal{D}}\ f(x) = \inf_{x \in ...
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33 views

Geometry question with convexity

Assume that a function $h(\lambda)$ is decreasing and convex given interval $[l,u]$ and has an unique root $\lambda^*\in (l,u)$. Also, assume $|l-\lambda^*| > |\lambda^*-u|$. Consider any $z\in ...
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54 views

Dual of a convex function $f:\mathbb{R} \to \mathbb{R}$: existence, solution to ODE

Let $f(x)$ be a smooth strictly convex (i.e. $f''(x)>0 \,\,\,\,\,\forall x)$ funtion of $x\in \mathbb{R}$. Define the dual function $F(p)$ as $$F(p)=\max_x [px-f(x)].$$ Make a sketch ...
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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 $Y$ ...
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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. ...
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Monotononically Increasing Water Filling Solution?

$\mathbf{I}$ is the $K\times K$ identity matrix. $\mathbf{h}_i\in\mathbb{C}^{M\times1}\quad\forall1\leq i\leq K$ are column vectors. Consider the solution of the convex optimisation problem over ...
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35 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 ...
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40 views

Finding the Expansion of a Separable Convex Optimization Problem

Hi there is a convex optimization problem in this paper which I am trying to implement in mosek. The author specifies that they also implement it using the separable optimization method. Specifically ...
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29 views

nonsurjectivity of Banach-Stone theorem

Currently I'm studying the extension of Banach-Stone theorem using this book. There is one section about the removal of surjectivity of the isometric isomorphism between the two function spaces $C(K)$ ...
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Convex Sets in Functional Analysis?

Why is it that convex sets and convex functions are a) so important & b) so intrinsically related to functional analysis as to deserve an entire chapter in Bourbaki's topological vector spaces? ...
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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 ...
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37 views

Probability that a $n$-dimensional Gaussian falls into a half-space

For $a \in \mathbb{R}_{\ge 0}^d$ and $b \in \mathbb{R}_{\ge 0}$, we can define a half-space as the set of points $x \in \mathbb{R}^d$ such that $a \cdot x \le b$, namely, $$\mathcal{H}(a,b) = \{x \in ...
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25 views

show this Polyhedral set is convex

show that polyhedral set is convex. A is a matrix ( m x n ) and b is element of Rm (i think)
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Convex set, quadratic form

I'm trying to answer a question concerning convex sets "Does the following constraint system define a convex set? $x^T Qx ≤ 1$ $a^T x = 0$ Here, Q is a symmetric and positive definite matrix and a ...
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27 views

Convex set, vector norms

I'm trying to solve the following question but I'm stuck. "Which of the following constraints define a convex set: ∥x∥ ≤ 1, ∥x∥ = 1, ∥x∥ ≥ 1?" The way to check for convexity is basically the same in ...
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intersection of two convex hulls

Let $X=\{x_1,\ldots,x_p\}\subseteq\mathbb{R}^n$ and $Y=\{y_1,\ldots,y_q\}\subseteq\mathbb{R}^n$. Is there a method (by using some algorithm) to find $\mathrm{conv}(X)\cap \mathrm{conv}(Y)$ as ...
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Is projection of a convex polyhedron on a plane a convex polygon?

If we have a convex polyhedron with vertices $\mathbf{V}$ and project it on a plane $\mathbf{P}$, is this procedure equivalent to projecting points in $\mathbf{V}$ on the plane $\mathbf{P}$ and then ...
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Convexity of a region on probability simplex

Exercise 2.15 g of Boyd et al Convex Optimization book : On the probability simplex in R^n where each point p = (p1,p2,p3...pn) corresponds to a distribution for random variable X to take one of n ...
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Shortest closed loop containing all extreme points of a convex set

Suppose $S\subset \mathbb{R}^2$ is compact and convex. Suppose $\Gamma:[0,1]\to\mathbb{R}^2$ is a continuous map with $\Gamma(0)=\Gamma(1)$. Suppose $\Gamma$ passes through all extreme points of $S$. ...
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Is linear function convex or concave?

I was wondering if linear function is convex or concave? For example f(x)=x, is function whose second derivate is 0 so we cant tell anything using this criteria. Can someone help?
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Existence of points in closed and bounded convex sets that cannot be expressed as convex combination of other elements of the set.

I have an intuition about convex, closed, bounded sets but I can't really find a way to prove whether it's right or wrong. Let $\Sigma$ be a convex set, that means, that given any $A,B \in \Sigma$, ...
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Is $g(x^n)$ a convex function of $x$, if $g$ is a convex function of $x$ and $n>2$; given $x$ is nonnegative?

Is $g(x^n)$ a convex function of $x$, if $g$ is a convex function of $x$ and $n>2$; given $x$ is nonnegative? like $g(x^2)$ or $g(x^3)$
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Conxex combinations of max and min

Is the following true? $$α \left( \max_{p\in P}\int g\mathrm dp\right)+\left (1-\alpha \right ) \left( \max_{q\in Q}\int g\mathrm dq \right )=\max_{z\in\left (\alpha P+\left(1-\alpha \right )Q ...
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What is the name of this object?

Suppose I have a convex set $K\subset X$, where $X$ is say a real Hilbert space (for simplicity). Then, given some $a\in \Bbb{R}$, let $$ \hat{K}=\{x:\langle x,y\rangle \leq a \;\forall y\in K\} $$ ...
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What is the motivation behind strong convexity

Definition : A function is said to be $\beta$-strongly convex if, $f(\theta w + (1-\theta) w') \le \theta f(w) + (1-\theta) f(w') - \frac{\beta}{2}\theta(1-\theta)(w-w')^2$ What is the motivation ...
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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 ...
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Does $\phi\circ g$ convex imply $\phi\circ f\circ g$ convex?

Assume that $\phi$ is an increasing function and $g$ is a decreasing function with $\phi\circ g$ convex. If $f$ is an increasing convex function does this imply $\phi\circ f\circ g$ is a convex ...
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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 ...
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64 views

Prove that this set is convex

Prove that $$ \left\{x=(x_1,x_2) \in \mathbb R^2 \mid \cos(x_1 + x_2) \ge \frac{\sqrt{2}}{2}, x_1^2 + x_2^2 \le \frac{\pi^2}{4}\right\}$$ is convex. How should I do this? Hessian is made out of ...
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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 ...
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Dominated Convergence on risk measures

This is a quite specific question and I am not able to provide the whole background (e.g. what a risk measure is). If someone knows that would be great. I am having difficulties understanding a ...
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44 views

Proving $x<y<z \implies \frac{f(y)-f(x)}{y-x} \le \frac{f(z)-f(y)}{z-y}$

Suppose $f$ is convex on $I$ and $(x,y,z)\in I^3$: How to prove that: $$x<y<z \implies \frac{f(y)-f(x)}{y-x} \le \frac{f(z)-f(y)}{z-y}$$
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Prove that the set is convex

$$x\in \Bbb R^2$$ $$4x_1^2 + 4x_2^2 \le 2x_1x_2 - x_1 + 2$$ I don't know how to prove that this set is convex, I can't find anything understandable either. The only thing I found is: $f(\theta x + ...
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Is this 0-sublevel $\sum (a_i 2^{\alpha_i x}) - \sum (b_i 2^{\beta_i x}) \leq 0$ a convex set?

I have this 0-sublevel set $a 2^{\alpha x} - b 2^{\beta x} \leq 0$ where $a_i$ and $b_i$ are non-negative. I can proof its convexity by using the definition. First, whenever $\alpha x$ and $\beta x$ ...