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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Is second derivative of a convex function convex?

If $f$ is twice differentiable and convex, is it true that $f''$ is a convex function ?
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1answer
23 views

Is mutual information convex in the joint distribution?

Assume some joint distribution $P(X,Y) = P(Y|X)P(X)$. It is well know that, for fixed $P(Y|X)$, mutual information is a concave function of $P(X)$ and, for fixed $P(X)$, a convex function of $P(Y|X)$ ...
12
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2answers
575 views

invex functions and their usefulness?

An invex function $f$ is a differentiable function from $\Bbb R^n$ to $\Bbb R$ that for some function $\eta : \Bbb R^n \times \Bbb R^n \to \Bbb R^n$ satisfies for all $x, u$, $f(x) - f(u) \geq \eta(x, ...
0
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1answer
11 views

Continuity of convex functions that have continuous restrictions to closed subspaces

Let $X$ be an infinite-dimensional normed vector space , let $U\subset X$ be an infinite-dimensional closed subspace, and let $f:X\to[0,\infty)$ be convex. Question: If the restriction $f|_U$ is ...
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0answers
49 views

Intersecting convex figures

Take in the real plane a finitely long horizontal line segment and connect the two endpoints by a convex path, above the segment, with the property that the only extreme points of the convex hull of ...
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0answers
47 views

Practical application of Gauss-Lucas theorem

Let $z_1,z_2,z_3 \in \mathbb C$ pairwise distinct be the affix of points $A, B$ and $C$. Let $P(x)=(x-z_1)(x-z_2)(x-z_3)$. Let $z_4$ and $z_5$ be the roots of $P'$ (with the possibilty that ...
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2answers
46 views

Is the function $\ln(ax + b)$ increasing/decreasing, concave/convex?

$h(x) = \ln(ax + b)$ NB. Examine your results acccording to values of $(a,b)$ I've differentiated twice in order to get the following: $$ h''(x) = -\frac{a^2}{(ax+b)^2} $$ I think this proves ...
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0answers
28 views

Upper bound using a convex function

Let $g, f: K\times S \to \Bbb R$ be convex and continuous functions on compact and convex sets $K,S \subset \Bbb R^n$. Does there exist a differentiable strongly convex function $h$ on $S$ (w.r.t. ...
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1answer
35 views

Complex analysis question regarding Cauchy's integral formula and holomorphic functions

Let $U \subseteq \Bbb{C} $ be an open convex subset of $\Bbb{C}$ We also assume that $\partial U$ is smooth. We fix a point $z_0 \in U.$ Using Cauchy's integral formula, show that $$\lvert ...
4
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1answer
36 views

Area of the lattice generated from $(n, n\sqrt{2} \mod 1)$

I plotted $\Big\{ (n, n \sqrt{2} \, \mathrm{mod} \,1) \;\Big| -50 \leq n \leq 50 \Big\}$ and even though the $n \sqrt{2}$ is a line, the pattern that emerges is a lattice. What is the basis of this ...
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1answer
30 views

Finding KKT conditions for nonlinear optimization problem.

I have an optimization like below: $\text{ minimize } \sum_k - \log_2 x_k $ $\text{subject to: } x_k \leq q , k =1,2, \cdots, N .$ I can form the Lagrange of the problem as below: $L(x, ...
1
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1answer
18 views

Convex hull of $3$ dimensional set reduced to $2$ dimensional set

Let $S = \{(f_1(t), f_2(t), f_3(t)) : t \in \mathbb{R}\}$ and suppose $f_3(t) \geq 1$ for all $t \in \mathbb{R}$. Is finding the convex hull of $S$ in some way equivalent to find the convex hull of $T ...
1
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1answer
27 views

Prove that all terms of a sequence of functions are convex.

Let $\ f_{n}: [0,1] \rightarrow \mathbb R, \quad f_{n}(x) = \left(e^{x}\right)^{1/n}.$ Is there a natural $n$ such that $f_{n}$ is concave on $[0,1]$? So second derivative is ...
1
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2answers
68 views

Prove that function is convex

Let $f\colon [a,b] \rightarrow \mathbb R$ be continuous and convex. Let $m \colon [a,b] \rightarrow \mathbb R$ and $m(x) = \max \left\{f(y): y \in [a,x] \right\}$. Prove that $m$ is convex I ...
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1answer
22 views

Dual Cone Construction $\{z \; | \;z \perp v \text{ for some } v \in \Lambda \}$

In a linear algebra computation, in order to estimate the second eigenvalue we consider a collection of vectors. Let $\Lambda$ be a cone in $\mathbb{R}^d$ then $$ \Lambda' = \Big\{z \;\Big| \;z ...
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0answers
20 views

Describing set of points where a convex function is differentiable

I've been told that the set of points at which a convex function $f: \mathbb R^n\rightarrow \mathbb R$ is differentiable is an $F_{\sigma}$ set, and I was hoping someone could help me see this. ...
3
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1answer
577 views

Proximal mapping for composition of functions

Suppose I have a convex function $f(x)$ for which I can easily compute the proximal mapping prox$_f(z) = \arg\min_{x} f(x) + \frac{1}{2}||x-z||^2_2$ is there a simple expression for the proximal ...
1
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0answers
18 views

Condition for product of increasing and decreasing functions to be quasiconcave?

Is there any condition for product of increasing and decreasing functions to be quasiconcave? More specifically, I am having in mind a condition for $F(x)\cdot(1-G(x))$ to be quasi concave where $F$ ...
0
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1answer
45 views

Line segment in the unit sphere

I want to prove the following statement Let $X$ be a normed linear space, with linearly independent vectors $x,y$, such that $\|x\|=1=\|y\|$, with $\|x\|+\|y\|=\|x+y\|$, then there is a line ...
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0answers
28 views

Proof of Separation Thm, intermediate step

This is an intermediate step required to prove the Separation Thm (https://en.wikipedia.org/wiki/Hyperplane_separation_theorem) Let $S,T\subseteq \mathbb{R}^n$ be nonempty, convex. Say ...
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1answer
36 views

Application of convex functions in economy [closed]

I have read in some texts that convex functions has application in economy. I want to see some clear examples of such applications.
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0answers
61 views

Optimizing over intersection of polytopes inside polytope

I have the following situation, as shown in the following diagram: $W=\{w_i\}_{i=1..|W|}$ is a set of vertices (show in diagram in blue) located within a regular simplex and having coordinates $\in ...
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1answer
21 views

Is the following function strongly convex?

I've tried to check the Hessian and see whether the following function is strongly convex w.r.t the euclidian norm however this function is not diff. at the origin. $g(x)=||x||_2+||x||_2^2$ Thanks
3
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1answer
174 views

Mid-point convexity does not imply convexity

A function $f: X \rightarrow \mathbb{R}$ is said to be mid-point convex if for all $x, y \in X$, we have $f(\frac{x + y}{2}) \leq \frac{f(x) + f(y)}{2}$. Can you please give an example of a function ...
0
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1answer
22 views

If every convex function is of bounded variation?

The properties of convex functions are of interest. I would like to know that if every convex function is of bounded variation?
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1answer
25 views

Convexity of Determinant of linear combination

Is it possible to show that the following is a convex function in $x$? $f(x)=\det(\sum_i x_i A_i)$ $A_i$ are real symmetric, positive definite matrices. Minkowski's inequality doesn't seem to do ...
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0answers
18 views

For polytopes, does union and linear transformation commute?

Given two (convex) polytopes $P_1$ and $P_2$ and a linear transformation $T$, is it true that: $$T(P_1 \cup P_2) = TP_1 \cup TP_2$$ What if $P_1$ and $P_2$ are not convex?
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101 views

Showing convexity of a function with the restriction over an arbitrary line

Let $f : \mathbb{R}^n → \mathbb{R}_∞$ be a function and let $C ⊂ dom f$ be a convex set. $$**Part I**$$ Prove that $f$ is a convex function if and only if $f$ is convex over every line $L_{v,x_0}$ ...
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1answer
256 views

Proof that a coordinate-wise convex function is convex?

I feel like this should be straightforward, but does anyone have a proof of the following? Let $f: \mathbb{R}^n \to \mathbb{R}$ satisfy the following. For each coordinate $i$, for an arbitrary vector ...
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0answers
16 views

Is this function strongly convex?

Let A,B be two intervals in $R$ and let $f(x,y):A\times B\rightarrow R $ be a continues function. Assume that $f$ is convex in both $x$ and $y$ Is the following function $g:A \rightarrow R$ is ...
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0answers
18 views

Positive hyperplane? Is there a name for these type of hyperplanes?

Consider an affine hyperplane $\{ x : \langle x,v\rangle=a \}$ where $a\ge 0$ and $v\in\mathbb{R}^n_{+}$. That is, both the level $a$ and the vector $v$ are non-negative. Is there any special name ...
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2answers
111 views

how to check an optimization function is convex or not

This is the sparse coding optimization function: $\operatorname*{argmin}_{B, \alpha} \sum_j \| \bf{x}_j - B\bf{\alpha}_j \|_2^2 + \lambda\sum_j |\bf{\alpha}_j|_1$ I read in the literature that this ...
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1answer
28 views

Notion of convexity on nonconvex domain?

Let $X\subset \mathbb{R}$ be potentially a nonconvex set, and let $f:X\rightarrow \mathbb{R}$ satisfy $\frac{f(x)-f(x')}{x-x'}$ increasing in $x$ and $x'$ for all $x,x'\in X$. Is there a name for this ...
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0answers
16 views

If a function is strongly convex on a set $X$, then is $f + \delta_X$ strongly convex on the closure of $X$?

If $f:\Omega \mapsto \mathbb{R}$ is strongly convex on $X \subset \Omega$, then can one say that $f + \delta_\bar{X}$ is strongly convex on $\bar{X}$? (Here $\bar{X}$ is the closure of $X$ and ...
3
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1answer
31 views

A question related to convex and compact set

I encounter a problem related to convex and compact set, which is stated as follows. Whether or not the following claim is correct? Claim: Let $C$ be an arbitrary subset of $R^n$ such that $C$ is ...
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0answers
28 views

Test for convexity

Consider the online learning setting where instantaneous loss is given by \begin{equation} \ell_t(f_t;(\mathbf{x}_t,y_t))=\max \left( {0,\left( \left( {\frac{N}{P}} \right){I_{(y_t = 1)}} + \left( ...
0
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1answer
36 views

convex bounded, closed $\Longrightarrow$ compact?

I know that A subset S of $\mathbb{R}^n$ is compact if it is bounded and closed (Heire-Borel theorem), Howver, if S is convex, containing the origin, closed, is it compact? Thanks in advance!
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1answer
363 views

Cones of positive semidefinite matrices generated by matrices of rank $1$

Let $S_n$ be the space of real $n \times n$ symmetric matrices and let $S_n^+$ be the convex cone of positive semidefinite matrices in $S_n$. The extremal rays of this cone correspond to the positive ...
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0answers
31 views

Least expensive way to “walk” with a convex potential

Let $V(x)$ be a non-decreasing, convex function on $\mathbb{R}^+$. Fix some positive integers $L$ and $k \leq L$. Let $A_{L,k}$ be the space whose elements are sequences of positive integers $r= ...
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0answers
37 views

Prove that the set of extreme points of a compact convex set is not empty.

The Krein–Milman theorem states that if $S$ is convex and compact in a locally convex space, then $S$ is the closed convex hull of its extreme points. In particular, such a set has extreme points. Is ...
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14 views

A metric space and polytopes

suppose we define a metric space $L_2(p)$ induced by a duality pair given by $\langle x,y\rangle =\sum_{j,k} p(j,k)[ x_1^j y_1^j+ x_2^k y_2^k]$ Where $x=(x_1,x_2)$ and $y=(y_1,y_2)$ and $p$ is a ...
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1answer
698 views

convex hull function in matlab

Is there any way to compute the convex hull of a finite set of points in Matlab and gives the half-space representation as its result? I usually use a toolbox called MPT developed at ETHZ Zurich, but ...
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0answers
13 views

Finding an unfrustrated set of local linear constraints with given minimal value

Let $ \{F_{i}\} $ be a finite set of linear functional on a convex closed subset $B\subset\mathbb{R}^{n}$ such that each $F_i$ is $k-local$ (acts on $k<<n$ variables only). Assume ...
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2answers
31 views

Is convex hull of a finite set a linear subspace of linear hull?

We have some convex and compact supset $G$ of Banach space $B$ and finitely many points ${x_1,...,x_N}$ . The question is : does the convex hull $C$ of ${x_1,...,x_N}$ a linear subspace of space ...
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2answers
157 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,\ ...
17
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1answer
319 views

Does there exist regular planar pentagon?

How to prove or disprove that the boundary of any convex body in $\mathbb{R}^3$ (treated as a surface) includes 5 points which form a regular planar pentagon?
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1answer
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One definition of strong convexity (from textbook of Prof. Bertsekas in 2015)

In strong convexity, there are a few definitions, one of them is: $f$ is strongly convex over $\mathcal{C}$ with coefficient $\sigma$ if $\forall x,y \in \mathcal{C}$ and all $\alpha \in [0,1]$, we ...
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0answers
103 views

Write a random variable as a convex combination of other 2

I'm trying to prove that if $f:[0,1]\to\mathbb{R} $ is continuous and convex, then the Bernstein polynomials are too. The hint that I've got is this: "Let $p_1 < p_2 < p_3<1$ and consider ...
6
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1answer
396 views

A convex function is differentiable at all but countably many points

Let $f:\Bbb R\to\Bbb R$ be a convex function. Then $f$ is differentiable at all but countably many points. It is clear that a convex function can be non-differentiable at countably many points, ...
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1answer
41 views

The maximum of several affine functions is a polyhedral function

A function $f: \mathbb{R}^n \mapsto (-\infty,\infty]$ is polyhedral if its epigraph is a polyhedral, i.e. $$\text{epi}f=\{(x,t)\in \mathbb{R}^{n+1} | \ \ C\left( \begin{matrix} x\\ t \end{matrix} ...