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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Image of convex hull

I came across a problem that I could simplify, if I knew that this is true: Let $A:= conv(x,y,z)$, where $x,y,z \in \mathbb{R}^n$ and $T: \mathbb{R}^n \rightarrow \mathbb{R}^n$ is a linear map. Does ...
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1answer
37 views

Is the biconjugate of a continuous functions also continuous?

Let $f:\mathbb{R}^n\to\mathbb{R}$ be given and assume that $|f(x)|\leq C|x|^2$. Is it true that the bi-(convex/Fenchel)-conjugate $f^{**}$ is also continuous. It was claimed in a book without a ...
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0answers
26 views

Intersections of convex hulls

Given a set of $n$ points $\{A_1, \ldots , A_n\}$ of the plane and every possible triangle formed with $3$ points $A$, I would like to describe the intersections fo theses triangles. By intersection, ...
2
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60 views

the continuity of argmin

how can I show the minimum of a converging sequence of continuous functions is converging to the minimum of the limit of that sequence.i.e. $$\lim_i \text{argmin}_{x} f_{i}(x)=\text{argmin}_{x} \lim_i ...
1
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1answer
54 views

Showing convexity proof

Let $F: \mathbb{R}^n \rightarrow \mathbb{R}^m$ be an affine function, i.e., $F (x) = L(x) + b$, with $L : \mathbb{R}^n \rightarrow \mathbb{R}^m$ linear and $b \in \mathbb{R}^m$ Then for every convex ...
0
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1answer
47 views

A condition for mid-convex implies convex

Let I an open interval, and $ f: I \rightarrow \mathbb{R} $ such that: $\forall (x,y) \in I^2 ; f(\frac{x+y}{2}) \le \frac{f(x)+f(y)}{2}$ There exists an interval $[a,b]$ such that $a<b$ and ...
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0answers
14 views

Is epi(max(f,g)) the intersection of epi(f) and epi(g)?

On an exam, I found the question "is max($f(x),g(x))$" convex if $f,g$ are convex? This lead me to the question in the topic. Is the intersection of epi$(f)$ and epi$(g)$ = epi($\max(f,g)$)? If so, ...
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1answer
17 views

About vertices of the convex hull of any finite set of points in $\mathbb R^n$

Let $S$ be a finite subset of $\mathbb R^n$ , we know that $x \in S$ is a vertex of $Conv (S)$ , the convex hull or convex polytope of $S$ , iff $x \notin Conv\Big(S$ \ $\{x\}\Big)$ ; then is the no. ...
2
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4answers
86 views

Sufficient condition for convexity

Let f:$ [a,b] \rightarrow \mathbb{R} $ a continous function such that $ \forall (x,y) \in [a,b]^{2}, \exists t \in ]0,1[, f(tx+(1-t)y) \le tf(x) + (1-t)f(y) $ show that f is convex
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44 views

Minkowski sum and difference do not cancel

I need to prove that $ ( A\oplus B )\ominus$ $B$ and $( A\ominus B)\oplus$ $B$ need not equal $A$ for all sets $A$, $B$,where $\oplus$ and $\ominus$ denote the Minkowski sum and difference. As ...
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1answer
37 views

How do I see that every point inside the corresponding convex region in $\mathbb R^2$ belong to this set?

Convex set in $\mathbb R^2$. Suppose I use the convex operator $\text {conv}$ to create the convex set of $X = \{x_1, ... , x_n\} \subset \mathbb R^2$, that is $\text {conv}(X) = \{(1-\lambda)x_i + ...
2
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1answer
57 views

What is the motivation behind the, convex and concave closures of submodular functions?

What is the motivation behind the , convex and concave closures of submodular functions? Also, my understanding is that the submodularity condition is somewhat like concavity which makes it counter ...
6
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2answers
83 views

How prove this $\frac{af(a)+bf(b)}{a+b}\ge f(a+b)$

Assume that $f(x)$ has two derivatives on $(0,2)$ and $0<a<b<a+b<2$. I have to prove that, if $f(a)\ge f(a+b)$ and $f''(x)\le 0$, then: $$\dfrac{af(a)+bf(b)}{a+b}\ge f(a+b).$$ I ...
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2answers
40 views

Function on convex set is convex if all rays are convex

Consider the function $f:D\rightarrow\mathbb{R}$ for $D\subset\mathbb{R}^n$ an open convex set. Furthermore, suppose that $g(t)=f(t\boldsymbol{x})$ is convex for all $\boldsymbol{x}\in D$. Is it ...
1
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1answer
20 views

Vector defined function is convex implies scalar defined function is convex

Let $f:\mathbb{R}^n \to \mathbb{R}$ be convex. Let $g:[0,1]\to \mathbb{R}, g(a)=f(a \cdot x+(1-a) \cdot y)$. Why does $f$-convex on $\mathbb{R}^n$ imply that $g$-convex on $[0,1]$?
2
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1answer
77 views

Convex function property

Let $ f_{1}, f_{2},..., f_{n} $ convex functions in the interval $[0,1]$ such that $ max(f_{1},f_{2},...,f_{n}) \geq 0 $ show that there exist positive real numbers $a_{1}, a_{2},...,a_{n} $ not ...
0
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1answer
43 views

Concavity of a multivariate function

Let f be a function such that f is Frechet differentiable. Prove that f is concave if and only if the following inequality holds: $$ 0\le ...
3
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0answers
54 views

Proving or disproving concavity of a function

I want to prove that the following function is concave (as a part of another proof). $$f(p) = \max_{\begin{matrix}x,y\\0\le x \le 1\\0\le y \le 1 \\ x * y = p\end{matrix}} \lambda h(x) + ...
1
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1answer
52 views

Representation of a point in terms of extreme points and extreme directions in a LP feasible region

We know that every point in a feasible region (X) of a Linear Program can be represented as a convex combination of the extreme points of X plus a non-negative combination of the extreme directions of ...
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2answers
95 views

Convex set with empty interior is nowhere dense?

Suppose $C\subseteq\mathbb R^n$ is a convex set and $C^o=\varnothing$. Is it necessarily true that $(\overline C)^o=\varnothing$? In general, is this true if $\mathbb R^n$ is replaced by a topological ...
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1answer
19 views

Conjectured characterization of a set relative to a convex cone

Let $X\subset \mathbb{R}^N$ be a convex cone (i.e., for all $x,y\in X$ and $\alpha,\beta\geq 0$ scalars, $\alpha x+\beta y\in X$). Define the set $$A(x)=\{a:x+a\in X \wedge x-a\in X\}.$$ Then, ...
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1answer
66 views

In finite dimensional normed space, every convex set contains a basis

I've been reading Lemma 5.60 here: http://epge.fgv.br/we/MD/TeoriaEconomicaAvancadaI/2009?action=AttachFile&do=get&target=Aliprantis-Infinite-Dimensional-Analysis.pdf (p.g 200, =217 on the ...
4
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1answer
109 views

Prove a certain property of linear functionals, using the Hahn-Banach-Separation theorems

I've been trying to solve this question, with no luck so far: Let $X$ be a real linear space, and $\{\|\cdot \|_i\}_{i=1}^{n}$ family of norms on $X$. Let $f$ be a linear functional on $X$ such that ...
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0answers
33 views

How nuclear norm is convex whereas weighted nuclaer norm is not?

In (http://nuit-blanche.blogspot.in/2014/05/wnnm-weighted-nuclear-norm-minimization.html), it is stated that nuclear norm of a matrix $\mathbf{X}$, given as $||\mathbf{X}||_{*}=\sum_{i} ...
0
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1answer
54 views

Convex conjugate of a function of sum of norms

I am trying to find the conjugate of function $f(x) = \|x\|_2 + \frac{1}{2} \|x\|_2^2$ i.e., $f^*(v) = \sup_x (v^Tx - f(x))$ where $x \in\mathbb R^n$ Although $f(x)$ is convex, I am stuck as the ...
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1answer
44 views

How to check convexity of a composition when some properties of inner and outer functions are known?

If $g(x)$ function is concave in $x$, and we want $g( f(x) )$ (where $f(x)$ is another function) to be convex in $x$, what are the required properties of $g(x)$ and $f(x)$? It would be appreciated ...
2
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1answer
67 views

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 \\ ...
0
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1answer
27 views

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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105 views

What are some examples of isotrophic sets?

What are some examples of isotrophic sets? and is there a "good" way to describe them? Isotrophic meaning that a random vector X uniformly distributed in the set has the isotrophic property for all ...
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34 views

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 $f$ ...
3
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0answers
81 views

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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81 views

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 ...
3
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2answers
43 views

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 ...
3
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0answers
114 views

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 ...
2
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1answer
63 views

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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2answers
38 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
2
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1answer
87 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 ...
3
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42 views

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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0answers
33 views

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. ...
2
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2answers
37 views

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 ...
3
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1answer
38 views

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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2answers
51 views

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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23 views

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 ...
0
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2answers
70 views

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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1answer
48 views

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 ...
4
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2answers
65 views

$\{x_n\} \to x$ 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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0answers
33 views

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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1answer
75 views

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 ...
1
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2answers
51 views

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 ...
4
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1answer
39 views

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) ...