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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Conditional expectation over a convex set

Let $\boldsymbol{X}$ be an $\mathbb{R}^d$-valued absolutely continuous and integrable random vector. Let $L \subset \mathbb{R}^d$ be a closed convex set. Does it hold that $\mathbb{E}[\boldsymbol{X} \ ...
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1answer
15 views

How can I compute fast the minimum of a linear plus Kulback-Leibler on the unit simplex?

Given $a, x^0 \in \mathbb{R}^n$ I wish to compute $$\min_{x \in \Delta_n} a^t x + \sum_{i=1}^n x_i\log(x_i/x^0_i) - x_i +x^0_i $$ where $\Delta_n$ is the unit simplex $\{x \in \mathbb{R}^n \mid ...
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1answer
34 views

$\lambda f(x)+(1-\lambda)f(y)=f(\lambda x+(1-\lambda)y)$ implies $f $ linear?

Let $X$ be a Banach space. $f:X \to X$ a continuous function. If we assume that $f$ satisfies the following convexity condition: $$\lambda f(x)+(1-\lambda)f(y)=f(\lambda x+(1-\lambda)y),$$ for all ...
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1answer
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Are unions, sums (…) of quasiconvex functions again quasiconvex?

for a project I need to prove quasiconvexity of several general functions. Can I argue that the union (or sum, or difference...) of quasiconvex functions is again quasiconvex? I do know that the sum ...
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1answer
20 views

Congruence Property of Monotone Operators

A map $T$ is called strictly monotone if for $x\ne y$, $\langle u-v,x-y\rangle>0$ for all $u\in T(x),v\in T(y)$. Let $A$ be an $m\times n$ matrix and $b\in\mathbb R^m$. I want to prove that if ...
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1answer
27 views

Prove that $|\int_C f(z)dz| \le M |z_2 - z_1|$ where $M \gt 0$ such that $|f(z)|\le M; \ \forall \ z \in \Omega$

Let $z_1$ and $z_2$ be any two points in $\Omega$ and let $C$ be any oriented contour in $\Omega$ from $z_1$ to $z_2$. Also, assume that $f:\Omega \to \Bbb{C}$ is analytic on an open convex set ...
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Can a non-convex set be partitioned into a set of nearly convex subsets?

Consider a non-convex bounded subset $S \subseteq \mathbb{R}^{n}$. Is it always possible to partition this set into a finite set of disjoint subsets \begin{equation} S = \bigcup_{i=1}^{n}s_i, \quad ...
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1answer
17 views

Is a distance function on $\mathbb{R}^n$ convex?

Fix $z \in \mathbb{R}^n$. Let $||\cdot||$ be a norm on $\mathbb{R}^n$, and define the distance function $f(x)=||z-x||$ for $x\in \mathbb{R}^n$ Then, is it true that $f(x)$ is convex?
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Proof of some inequality on convex sets in $\mathbb{R}^n$

Suppose $\Omega$ is a strictly convex bounded set with smooth boundary in $\mathbb{R}^n$. If $y \in \partial\Omega$, how can we show that $$\langle(x-y),\nu_y\rangle < 0$$ for all $x \in ...
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20 views

Specific polygons in \R^{3} [on hold]

Find all (convex) polyhedra P in $\mathbb{R}^{3}$ with following property: for every two vertices $v, u \in P$: $[u,v] \in \partial P$ . Here $[u, v] = \{w \; \vert \; w = tv + (1-t)u, \; t \in ...
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37 views

definition of largest circle containing in a convex body

For a convex body B(compact convex set with non-empty interior) what does each of these following values mean? 1- $$\ \sup_{x\in B}\inf_{y\in cB} d(x,y) $$ and 2- $$\ \sup_{x\in B}\inf_{y\in ...
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53 views

does this convex set have a specific name?

Let $x_1,\dots,x_N$ be points of $\mathbb{R}^n$. Define the following set: $\mathcal{A} = \left\{\sum_{j=1}^N a_j x_j : -1 \le a_j \le 1, \, \, \forall j=1,...,N\right\}$. It is an easy exercise to ...
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2answers
23 views

Uniformly convex and strictly convex

I have the following definitions of uniformly convex and strongly convex Let $f:R^n \to R$ be smooth. (1) $f$ is uniformly convex if there exists $\theta > 0 $ such that $$\Sigma_{i,j}f_{x_i x_j} ...
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1answer
7 views

Supporting hyperplane of convex function

Below is the appendix B of Evan's PDE book on supporting hyperplanes of convex functions. In the remark (1), he says that the mapping $y\to f(x)+r\cdot(y-x)$ determines the supporting hyperplane to ...
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Pos of Matrix is a closed set? [on hold]

"Pos of matrix" $:= \{ \sum_{i=1}^n \lambda_i *a_i \mid \lambda_i \geq 0 ,\ a_i \text{ are columns of matrix, } 1\leq i \leq n\}$ that matrix have $n$ columns. Prove that "Pos of matrix" is a closed ...
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20 views

pos of matrix is close [on hold]

"Pos of matrix" := $\{ \sum_{i=1}^{m}\lambda_i *a_i |\ \lambda_i \geq 0 ,\ a_i\ \text{are columns of matrix,}\ 1\leq i \leq n\}$ that Matrix have $n$ columns. prove that "Pos of matrix" ...
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1answer
20 views

Parametrizing the Boundary of a Convex Set

Let $K$ be a compact convex set in $\mathbb{R}^2$. In the proof of a proposition in a paper I am reading, they are concerned with parameterizing $\partial K$ in the following way: If $K$ is ...
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14 views

How can I find discrete points in convex hull

Say, I have a set of finite numbers of data $S = \{z_1, z_2,...,z_n\}$, $z_i \in \mathbb{Z}^d$, $C$ is the convex hull generated by $S$, that is, $C = conv(S)$. How can I find some discrete points in ...
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1answer
17 views

Showing that a given function is convex.

I am trying to show that the function $f(x,\vec{y})=\alpha\ln(1+\exp(x+\vec{y}\cdot \vec{z}))+(1-\alpha)\ln(1+\exp(-x-\vec{y}\cdot\vec{z}))$ is a convex function of $(x,\vec{y})$ (where ...
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1answer
43 views

Non-trivial lower bound approximation of a convex function using the second derivative at the minimum

Say that I am given an infinitely differentiable convex function $f: \mathbb{R}^n \rightarrow \mathbb{R}$. I am wondering if I can construct a meaningful lower bound approximation of $f$ using it's ...
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1answer
32 views

How to show that there's a continuous function separating convex sets of Radon measures?

First, the setup: $X$ is a compact set. By Riesz's representation theorem $C(X)^*=${all Radon measures on $X$}. $K$ is a convex, closed (weak-* sense) set of probability measures. $m$ is a probability ...
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+50

Relationships between affine closures and convex closures

Let $V$ denote a vector space. Then the following concepts make sense: affine subset of $V$ affine closure (affine "hull") of a subset of $V$ Suppose $V$ is in fact a real vector space. Then the ...
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33 views

Generalizing convexity of sets

Let $X$ be a subset of some Euclidean space. We say that $X$ is convex if for any two points $p$, $q$ in $X$, the line segment joining $p$ and $q$ is also in $X$. But what if we loosen this definition ...
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35 views

Application of the Operator norm $\|.\|_O$ on the differential $df \in \hom(\mathbb{R}^n, \mathbb{R}^m)$

This question origins from my Analysis II Script which gives the following statement (without proof): Lemma Let $U \subset \mathbb{R}^n$ be convex and $f \in C^1(U, \mathbb{R}^k)$ then we have $$ ...
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1answer
25 views

Strong convergence of convex combinations of a weakly convergent sequence

Consider the Mazur's Lemma (H. Brezis - "Functional analysis, ..."): Assume $(x_n)$ converges weakly to $x$. Then there exists a sequence $(y_n)$ made up of convex combinations of the $x_n$'s that ...
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does convexity implies $g(u)>cu$?

So I have been doing some self study and I was wondering if my results are true, or if I am misreading something. Say we have a function $g$ which is concave on values of $u \in \mathbb{R}$. Then we ...
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1answer
19 views

convexity and first derivative

Let $\phi$ be a differentiable function on an interval $(a,b)\subset R^1$. If $\phi '$ is non-decreasing, then $\phi$ is convex. But, is the converse true? Does the convexity of $\phi$ necessarily ...
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Takahashi convex metric spaces

A Takahashi convex metric space is a metric space $(X,d)$ such that $\exists W : X \times X \times [0,1] \rightarrow X$ that satisfies : $d (u, W(x,y; \lambda)) \leq \lambda d(u,x) + (1- \lambda) ...
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1answer
19 views

Show the Gini Coefficient is Quasiconvex

The Gini-coefficient is defined as $$ G(x) = \sum_{i = 1}^n \frac{i}{n} - \sum_{j=1}^{i} \frac{x_{(j)}}{\mathbb{1}^{T}x}, $$ where $x_{i} $ is nonnegative numbers with positive sum. $x_{(j)}$ denotes ...
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About the hyperplane conjecture.

I have recently heard about the hyperplane conjecture and I would like to understand better the problematic behind this conjecture. The hyperplane conjecture: There exists a universal constant ...
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29 views

To show $f(x)$ has ONLY one Max in $x\in[0,1]$

I have function $$f(x)=\left(\frac{1-x}{2-x}\right)x^{p-1}~\text{where}~p>1;~x\in[0,1]$$ I want to show that $f(x)$ has ONLY one maximum in $x\in[0,1]$. I get the second derivative as ...
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1answer
34 views

Show that f(x) is convex

Show that f(x) = inf{g(x1)+h(x2)} is convex subject to x1+x2=x and where g(.) and h(.) are convex functions. Can I just go about this by using the regular definition of a convex function or which ...
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1answer
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Convexity/concavity of a strictly increasing and continuous function

Consider a continuous, strictly increasing function $f:\mathbb{R}_{+}\to \mathbb{R}_{+}$ with $f(0)=0$, and $x>f(x)$ for all $x>0$. Is this enough to conclude anything about ...
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2answers
20 views

Finding when a function is concave [closed]

Given the function below $f=e^{ax^2+bx}$ what are the conditions on $a$ and $b$ such that $f$ wil be concave Thanks
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1answer
30 views

Are these two definitions of an affine subspace equivalent?

I've seen the notion of an affine subspace defined differently as follows: $S \subset \mathbb R^3$, non-empty, is an affine subspace if $(1-t)u + tv \in S$ whenever $u,v \in S$. $S$ is an affine ...
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strong convex implies exp-concave

Prove that if f is strong convex (for some m>0) $\mbox(\nabla f(\mathbf{x})-\nabla f(\mathbf{y}))^{T}(\mathbf{x}-\mathbf{y})\geq m||\mathbf{x}-\mathbf{y}||_{2}^{2} $ then f is also ...
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2answers
34 views

compute the smallest affine subspace containing $S$, where $S=\{(1,1,1),(2,3,4),(1,2,3),(2,1,0)\}$ is a set of vectors in $\mathbb R^3$

I've started to study convexity to enchance my optimization skills. Given a set $S=\{(1,1,1),(2,3,4),(1,2,3),(2,1,0)\}$ of vectors in $\mathbb R^3$ an exercise asks to compute the smallest affine ...
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1answer
48 views

Identity of the Operator Norm applied to the differential $df$ for convex $U \subset \mathbb{R}^n$ and $f \in C^1(U, \mathbb{R}^k)$

In my Analysis II Script they often use 'special' norms such as the Operator norm to make proofs more 'elegant' or just shorter. There is also the following statement (without a proof) which I can't ...
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converting a equation to convex form which can be given to cvx solver to solve it. [closed]

Can anybody tell me how to convert this to quadratic programming format so that CVX could solve it...?? I am not asking the whole solution but need only conversion. objective is:- minimize {sum ( ...
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Tangent plane of a convex set.

Given a convex set $\Omega$ in $\mathbb{R}^n$ with smooth boundary, is $\Omega$ contained completely on one side of it's tangent plane at any point on the boundary? If so, how can we prove this? ...
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1answer
16 views

Extending a convex function

Suppose $f:(a,b) \to \mathbb R$ is twice differentiable with the property that $c_1 \leq f''(x) \leq c_2$ for every $x \in (a,b)$, where $c_1, c_2$ are positive constants. Is it possible to extend $f$ ...
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4answers
75 views

Euclidean norm second derivative

I really need Your help. I need to prove that Euclidean norm is strictly convex. I know that a function is strictly convex if $f ''(x)>0$. Can I use it for Euclidean norm and how? ...
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1answer
506 views

proximal operator of infinity norm

What is the proximal operator of $\|x\|_\infty $? I know we have to take the subgradient and compute it but I am a bit stuck. Can anyone show me steps?
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1answer
35 views

Log-convexity preserved by sum?

I'm currently reading Boyd's book on Convex Optimization (free book which can be found here : http://stanford.edu/~boyd/cvxbook/) , and there is one proof I do not understand : p. 105, the book says ...
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1answer
30 views

Is this function of matrix F convex?

A scalar function $f(F)=vec(F)^{H}\Big(\Omega^{H}\big(M\otimes F(I+F^{H}F)^{-1}F^{H}\big)\Omega\Big)^{-1}vec(F)$ convex? $\Omega$ is arbitrary matrix and $M$ is positive definite. F can be reviewed ...
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3answers
985 views

convexity of inverse function

I have a question on the reverse of a convex function. Let $f(x)$ be a convex function. Is the reverse function, say $g(x)=f(x)^{-1}$, is necessarily a concave function ? Considering that such ...
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1answer
30 views

Are the coefficients of a Taylor series bounded when the function is?

Say that I have three real functions $f(x)$, $g(x)$, and $h(x)$ such that $f(x)\le g(x)\le h(x)$ for all real $x$. Additionally, $f(x)$ and $h(x)$ are logarithmically convex. Can I make any definite ...
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3answers
133 views

Is $f(x_1,x_2)=x_1x_2$, with $x_1,x_2 \in \mathbb{R}_+$, a convex function?

Let $f(x_1,x_2)=x_1x_2$, with $x_1,x_2 \ge 0$. Is $f$ a convex function? Why? $\ \\$ Edit (in view of the comments below) The Hessian matrix is $H=[0\, 1; \,1 \,0]$, which is indefinite (in ...
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2answers
60 views

Can $\sin (x)$ be represented as difference of two convex functions?

In my optimization homework, I am supposed to prove that every differentiable function with Lipschitz continuous gradients can be represented as difference of two convex functions. I think I have come ...
2
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1answer
35 views

Pointwise convergence implies uniform convergence under concavity?

Suppose that $X$ is a subset of a Euclidean space and $f_n:X\to\mathbb{R}$ converges pointwise to $f:X\to\mathbb{R}$. A book I'm reading seems to say that: if $X$ is convex and compact and each ...