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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Decomposing convex set into slices

Let $\Omega$ be a convex quadrilateral with vertices $a_1, a_2, a_3, a_4 \in R^2$. Can it be presented as the union of lines connecting the opposite edges $a_4-a_1$ and $a_3-a_2$ as follows? ...
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Normal Cone of $\mathbb{R}^n_+$ and $S^n$?

I'm trying to solve the problem $\min_x \{f(x) + \delta_X(x)\}$ where $f$ is a differentiable function and $\delta$ is the indicator function $\delta_X(x) = \begin{array}{l}0, x \in X \\ ...
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Eliminating equality constains

The following text derived from book convex optimization by Boyd, page 143. For a convex problem the equality constraints must be linear, i.e., of the form $Ax = b$. In this case they can be ...
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What are some examples of isotrophic sets?

What are some examples of isotrophic sets? and is there a "good" way to describe them?
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Surjective bilinear map

Let $Q$ be a CONVEX quadrilateral in $R^2$ with vertices $a_1,a_2,a_3,a_4 \in R^2$. Consider the bilinear map $f: [0,1]^2 \to Q$ $$f(x,y)=a_1+(a_2−a_1)x+(a_4−a_1)y+(a_1+a_3−a_2−a_4)xy$$ Note that $f$ ...
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Approximate the unit ball in an infinite-dimensional Hilbert space, by compact sets?

Are there some common ways to approximate the unit ball in an infinite-dimensional Hilbert space, by compact sets? (note that the unit ball isn't compact.) My goal is to prove a statement which holds ...
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Continuous functions of minimal norm

Let $C$ denote the set of continuos functions on $[0,1]$ with the supremum norm. $M\subset C$ such that $$\displaystyle\int_{0}^{1/2}f(t)\, dt-\int_{1/2}^{1}f(t)\, dt=1,\; \forall f\in M$$ Show ...
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Convex basis and conical basis (how to draw?)

There is a question, I'm struggling with: Find vertices of the following described polyhedron, $P:=P(A,b)=conv(V)+cone(E)$ where $V$ is the set of all vertices of $P$ and $E$ is the set of all ...
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Intersection of compact convexes

Let $C_1,C_2,C_3,C_4$ be compact convexes of $\mathbb{R}^2$ such that $C_1\cap C_2\cap C_3\neq\emptyset,C_1\cap C_2\cap C_4\neq\emptyset,C_1\cap C_3\cap C_4\neq\emptyset,C_2\cap C_3\cap ...
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A conjectural inequality of the form $\int_a^b g(B'_1(t)) dt \le \int_a^b g(B'_2(t)) dt $ with convex increasing $g$

Assume $ g:[0,\infty)\to [0,\infty) $ be strictly convex and increasing monotonic function and $B_1:[0,\infty)\to [0,\infty) $ be convex and increasing monotonic function and $B_2:[0,\infty)\to ...
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Show that the set $\{x\in\mathbb{R}^N:\nabla f(x)=0 \}$ is convex

Let $f:\mathbb{R}^N\rightarrow \mathbb{R}$ be a $C^1$ convex function. Show that $\{x\in\mathbb{R}^N:\nabla f(x)=0 \}$ is convex (we assume that empty set is convex). Any hint?
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34 views

Convex function and expectation

I was wondering: if f is a convex function and X a random variable, what does E(f(X)) = f(E(X)) implies? Thanks a lot, David
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50 views

Proving that quadratic form is convex in (vector, matrix) arguments

I'm studying with the quadratic form $$ F( (x,Q) ) = \langle x,Q^{-1}x\rangle $$ considered over $\mathbb{R}^n\times\mathbb{R}^{n\times n}_+$, where $\mathbb{R}^{n\times n}_+$ is the set of all ...
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Existence and uniqueness of a function generalizing a finite sum of powers of logarithms

I hope to find a proof of the following conjecture: $(1)$ For every $a>0$ there is a convex analytic function $f_a:\mathbb R^+\to\mathbb R$ such that: $f(1)=0$ and $\forall x>1,\ ...
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Is the ratio trace problem convex?

I have a ratio trace problem described as follows: $\arg\max_{w} trace((w^tAw)*inv(w^tBw))$, where A and B are full rank matrices. This problem can be solved via generalized eigenvalue problem. ...
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Splitting the plane to fit convexes

I'm trying to show the following : Let $K,L$ two closed convexes of $\mathbb{R}^2,O=(0,0)$ If $O\notin K$ then there exists a straight line $D$ going through $O$ such that $K$ is in one of the half ...
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A problem with equality in a inequality for convex function

Let $f:\rightarrow \mathbb R$ be a convex function on a convex subset $D$ of linear space $X$. Assume that for some pairwise disjoit $x_1,x_2,x_3\in D$ and some $t_1,t_2,t_3\in (0,1)$ such that ...
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what is the convex hull of such a matrix cone?

A matrix cone is in the following form: $M: = \begin{pmatrix} 1 \\ x\end{pmatrix}\begin{pmatrix}1 & x^T\end{pmatrix}$ where $x\in F$ , let $F = \{x: x\in [l,u]^n\}$ How to express the convex ...
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Convex ball in $L^p$ spaces

If $1<p<\infty$ prove that the unit ball of $L^p$ is strictly convex; this means that if $$\|f\|_p=\|g\|_p=1, \ \ f\neq g, \ \ \ h=\frac{1}{2}(f+g)$$ then $\|h\|_p<1$. By Minkowsky ...
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closed convex set, unique point that minimizes distance

Let $E \subset \mathbb{R}^k$ be a closed convex set. How would I go about showing that for each $x \in \mathbb{R}^k$ there is a unique $p \in E$ such that $|x-p| = \inf_{y \in E} |x - y|$?
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How to determine if the given points form a convex irregular Hexagon.

Say I have a collection of points (x,y). From the given points, I want to determine if it forms a convex irregular Hexagon. My goal is to determine that the points I have gathered form an irregular ...
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$\{x_n\} \to x$ iff $\bigcap_{n=1}^\infty K_n = \{x\}$

Let $\{x_n\}$ be a sequence in $\mathbb{R}^k$ and let $K_n$ be the intersection of all closed convex sets that contain $x_m$ for all $m \ge n$. How do I show that $\{x_n\}$ converges to $x$ if and ...
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Write a point on/inside the convex hull of a set of vertices

We work over $\mathbb{R}^N$. We consider a convex hull defined by its $L$ vertices. How to write a point $\textbf{on}$ (in) the convex hull in function of all the vertices? How to write a point ...
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Show that $y$ is convex in $x$ provided that $y=h(I)$ and $x=g(I)$

Suppose that $\phi:[0,\infty)\to[0,1]$ is strictly increasing, infinitely differentiable such that $I\mapsto(1-\phi(I))I$ is injective. Define $$ y=\phi(I)I,\quad x=(1-\phi(I))I. $$ I would ...
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Variation of Jensen-Inequality

I just read a variation of Jensen's Inequality which states: If $f: \mathbb{R} \rightarrow \mathbb{R} $ is a convex function, $ \phi \in \mathcal{L}^1(\mathbb{R}^n)$ with $ \phi \geq 0$ and $ \int ...
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Is $f(x)x$ convex for increasing function $f$?

Suppose that $f:(0,\infty)\to[0,1]$ is strictly increasing and infinitely differentiable. I have an intuition that $$ g(x)=f(x)x $$ should be convex in $x$ (i.e. increasing at accelerating rate) ...
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Is the composition of a set of convex functions convex?

Here we see the proof for $f(x)$ being convex where $$f(x) = h(g(x))$$given $h$ is convex and nondecreasing and $g$ is convex. But what if $$f(x) = h(g_1(x),g_2(x),g_3(x),...,g_k(x))$$ where each ...
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geometric interpretation of analytical hahn-banach theorem

I understand this interpretation. But how can I see this in example?
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Convex hull of finite union

Suppsose $A\subset\mathbb{R}^n$ and let co$(A)$ be its convex hull. Then does the following hold for $A_1,\cdots,A_k$: $\text{co}(A_1\cup\cdots\cup ...
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How to prove that a cone is closed?

How to prove a cone $K$ is closed ? I know that $K$ is a set, for a set, if it is not open, then it is closed. But how to prove that it is closed directly ?
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Prove this function is convex

we have $ g: R^n \rightarrow R$ is a concave function and $S$={$x :g(x)> 0$} and $f:S \rightarrow R$ and $f(x)$=$1/g(x)$ so we must show that $f$ is a convex function
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Draw a convex hull in 2D

What does the convex hull look like for the points below? is that just these three points connected by lines between these points? $\text{convexhull}\{(3,2);(2,3);(2.1,2.1)\}$ And can I draw this in ...
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Greatest norm for the exponential of a polytopic matrix

im new to the forum and I have a question that Im working quite for a while now. I would like to prove that the greatest norm when considering the exponential ...
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MVT and functions

Let $f$ be defined on an open interval $I := (a,b)$. (a) Let $x$ and $y$ be real numbers such that $x<y$. Show that if $z \in [x,y]$, then there is some $t \in [0,1]$ such that $z=tx+(1-t)y$. (b) ...
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Will this optimisation give the global maximum?

My book optimises a type of functions using the lagrange method. From calculus I remember that we had to check the boundary when using lagrange, because it only gave local max, but it is not mentioned ...
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Phase retrieval via SDP (semidefinite program) of 2D test image (Matrix completion)

In SDP based phase retrieval we have intensity measurement (abs(FFT2(x))^2) of the form A(xo) = |ak,x|^2 =b^2, phase retrieval is then find x that obeys A(xo)=b The quadratic measurements can be ...
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Closedness of convex sets in a locally convex space

Let $C$ be a convex subset of a locally convex topological vector space. Consider the properties: a) $C$ is closed. b) $C$ is weakly closed. c) $C$ is weakly sequentially closed. d) $C$ is ...
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Some properties of a random variable

I have absolutely no idea how to show this: Let $X$ be a random variable whose distribution is not degenerate. By considering the function $F( \theta) = \mathbb{E} U( \theta X)$, $\theta \in ...
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Is there a proof of subdifferential sum rule that doesn't use duality theory?

Given: $f$ and $g$ are lower-semicontinuous proper convex functions, $x \in \text{ri dom}(f) \cap \text{ri dom}(g)$, $h = f+g$, $p \in \partial h(x)$, Prove that there exist some $s \in \partial ...
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Lower semicontinuity vs closed sublevel sets

Let $V$ be a real locally convex space. Let $F : V \to R$. Are the following equivalent? (a) $\{ u \in V : F(u) \leq a \}$ is closed for any $a \in R$. (b) $\liminf_{n} F(u_n) \geq F(u)$ whenever ...
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Is the following convex geometry relating intersection and set averages true?

Let $X$ and $Y$ be two convex cones, and denote by $(1/2)*X+(1/2)*Y$ the Minkowski average of $X$ and $Y$ (i.e., $\{z:z=(1/2)*x+(1/2)*y,x\in X,y\in Y\}$). Then $$X\cap Y \subseteq(1/2)*X+(1/2)*Y$$ Is ...
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the derivative of a sequence of convex function coverge

Continue of the discusstion in: Limit of derivatives of convex functions It proves that: Let $(f_n)_ {n\in\mathbb{N}}$ be a sequence of convex differentiable functions on $\mathbb{R}$. Suppose that ...
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Lovasz Extension of the Product of Functions

Let $f$ and $g$ be submodular functions, and let $\widehat{f}$ and $\widehat{g}$ be the Lovasz extensions of $f$ and $g$, respectively. What can we say about the Lovasz extension of $f \times g$, ...
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continuous extensions of concave functions

Let $N$ be a lattice. For a ring R we denote $N_R := N \otimes R$. My question is the following: Does a continuous and concave function \begin{eqnarray*} f: N_{\mathbb{Q}} \to \mathbb{R} ...
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Largest and smallest shape enclosed within circles

There is a convex shape $C$. It is known that the largest disc contained in $C$ has radius $r$ and the smallest disc containing $C$ has radius $R$. What are the smallest-area and largest-area shapes ...
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How to show convexity of an inequality?

How would $x^2 + y^4 + z^2 < 5$ be shown to be convex? What I was thinking was that this is sort of like an ellipsoid and these are convex sets. The traditional definition of convexity for some ...
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what can be said about the solution of the following differential equation

$y(x)=f(x)+\log(1+y'(x))$ with $\lim_{x \to 0}f(x)=\infty,f(\infty)=0$ and $f(x)$ is contiuoues and decreasing. In particular, can we prove that the solution $y(x)$ is convex in general? from ...
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Find those values 'a' which belongs to the Convex Hull

Find those values of 'a' for which (1,a,1) belongs to the convex hull of $$\{(0,0,0), (1,1,2),(2,4,-6), (1,3,8)\}$$ Give me hints as much as you can, I would like to understand the mindset rather ...
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32 views

Conditions for unique solution of a maximization problem?

Let $S\subseteq \mathbb{R}^2$, $d:=(d_1,d_2) \in S$, and $s:=(s_1,s_2)$ a generic point of $S$. Assume that there exists $s \in S$ such that $s_1>d_1$ and $s_2 >d_2$. Consider the following ...
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43 views

Convex analysis question involving salient convex cones

Suppose $X$ is a salient convex cone. That is, if $x,y$$\in$$X$ and $\alpha,\beta$$\geq$$0$ are scalars, then $\alpha$$x$+$\beta$$y$$\in$$X$ and if $0\neq x$$\in$$X$, then $-x$ is not. Then for any ...