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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show that function is convex

Let $f:\mathbb{R}\to\overline{\mathbb{R}}$. Show that $$f\left(x\right)=\begin{cases} +\infty & \mbox{ if }x\in\left(0,\infty\right)\\ 0 & \mbox{ if }x=0\\ -\infty & \mbox{ if ...
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Measure of a set

Let $A \subset R$ such that for all open interval I, $m^* (A \cap I) < 1/2 L(I)$, where L is the length of a interval and $m^*$ is measure, prove that $m^*(A)=0$. I appreciate any hint to solve ...
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Finding Borel measures on a closed convex hull

Let $M=C(I)^{\ast}$, the space of complex Borel measures on the unit interval $I$. Suppose we give $M$ the weak*-topology induced by the Banach space $C(I)$. Now $\forall$ $t \in I$, let $e_t \in M$ ...
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Second derivative positive $\implies$ convex

In proof of the following theorem; If $f$ has a second derivative that is non-negative (positive) over an interval then $f$ is convex (strictly convex). $f$ is in real number space., the book I ...
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Polytopes inside polytopes

I would like to share some facts about polytopes and how to check whether a polytope is a subset of some other polytope. I will provide some methods and I would like to know whether there exist other ...
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167 views

Subsets of R2 that are convex, closed, and have non-empty interiors?

Can someone give me some guidance with this problem? Thanks. Suppose that $A, B \subset \mathbb{R}$ are convex, closed, and have non-empty interiors. Prove that $A, B$ are the closure of their ...
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1answer
70 views

How can we ensure that a space is a subset of locally convex topological space?

I am looking for fast ways to ensure that a given set is a subset of topologically locally convex space. I have already read the posts post1:seminorms-in-locally-convex-spaces, ...
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Strong convexity and Lipschitz

What can you say about the L and $\lambda$ for a $\lambda$-strongly convex differentiable function, if its gradient if L-Lipschitz? Also, it is given that $\lVert \nabla f(y) - \nabla f(x)\rVert_2 ...
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Maximizing a ratio of convex matrix functions by minimizing a difference?

Given that $g(.),h(.)$ are twice-differentiable convex quadratic real functions whose domain is the set of all real matrices while the range is the set of positive real numbers, then: Is maximizing ...
2
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245 views

Sum of Two Convex Sets

A friend of mine recently got an assignment, which asked for the sum of two convex sets in $\mathbb{R}^n$. Is this sum referring to Minkowski addition or is there another meaning to it? (such as the ...
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Showing existence of solution by positive definiteness/convexity

For a physics problem, I am considering the following problem: I have a certain function, $S: \mathbb{R}^M \rightarrow \mathbb{R}$, of which the critical points, given by $$ \frac{\partial ...
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Biggest ellipse included in a convex polygon

Considering a N edges convex 2D polygon called P. Let's name its vertices $\{p_1, p_2, ..., p_N\}$ described in a counter-clockwise order, with $p_i = (x_i, y_i)$ What would be, and how would one ...
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1answer
41 views

Is this a convext set?

Is this one a convex set? how to prove it? I failed to prove it through the definition of convex set. Thank you. $$\left\{(x_1,x_2)\mid\sqrt{x_1^2+x_2^2}+|x_1|+|x_2|\le 1\right\}$$
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294 views

Relationship between convexity and superadditivity?

This question is a little vague, so let me give some motivation. I was trying to prove the generalized Holder's inequality for probability measures, $$\mathbb{E}(X_1 \dots X_n) \leq \prod_{i=1}^n ...
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42 views

Convexity wrt a matrix, convexity wrt vectors

If a function $f(X)$ is a convex function of matrix $X$, does it imply that $f$ is also a convex function of all rows of $X$? (My final goal is to see if I can use coordinate descent by optimizing a ...
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81 views

Average in an open set, does it imply convexity?

Let $C$ be an open subset of $\mathbb{R}^n$. If for all $a,b \in C$, $(a+b)/2 \in C$, then prove that $C$ is convex.
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Explanation in some part of proof

It might be too specific to be asked here, but I can't figure out myself. I have this thm and my problem is in the necessity part: $$ \text{M is hyperplane iff } \exists h\in\mathbf{R}^n \text{ and } ...
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proof of convex set problem

I had a homework problem which asks to prove or disprove that A is a convex set where $A=\{x: g(x) \le c\}$. and $g(x)$ is a convex function. I went ahead in this way: Assume $x_1$ and $x_2$ $\in$ A. ...
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The most efficient algorithm to solve the following problem

Is there an efficient optimization algorithm to solve the following problem? $(\alpha,\beta,\gamma,\cdots) =$ argmax $\sum_{i}\log(\alpha a_i+\beta b_i+\gamma c_i+\cdots)$, s.t. ...
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1answer
152 views

KKT Conditions for Minmax Problem

Let $\mathbf{x}\in\mathbb{R}^n$ and $\mathbf{y}\in\mathbb{R}^m$. Now $$f\left(\mathbf{x}, \mathbf{y}\right):\mathbb{R}^n\times\mathbb{R}^m\rightarrow\mathbb{R}$$ is convex in $\mathbf{x}$ and concave ...
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Prove a function is non convex

I have a simple function dependent on two variables $x_1$ and $x_2$: $$ f(x) = \ln\left(\frac{x_1}{x_2}\right) $$ where $x_1, x_2 > 0$ (strictly positive). I know this function is non convex as, ...
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446 views

How to prove that the product of a decreasing monotonic function and a strictly increasing monotonic function is a concave function?

Given that I have a strictly increasing monotonic function $f$ and a decreasing monotonic function $g$, are there any nice properties to show that the product function $h(x) = f(x)g(x)$ is a concave ...
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1answer
824 views

Convexity of sum and intersection of convex sets

Let $A_i$ be a subset of $\Bbb{R}^m$ which is convex for $i=1,...,n$. How can I prove that the sum of $A_i$ is also convex? I know how to prove it with two sets: Let $x = a_1 + b_1$ and $y = a_2 ...
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$S$ convex $\implies f(S)$ convex

Trying to prove this since no optimization book in my hands proves it. The problem is that I know nothing about $f$. Here is my pathetic attempt Since S is convex, then $tx + (1 - t)y \in S$ for $t ...
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How many methods could be used to solve this optimization problem with equality constraints?

I wonder whether there is a simplest method for this problem. The function to maximize is $F(x)$. $F(x)=\|Kx\|_2^2=x^TK^TKx$, where $K\in \mathbb{R}^{n\times d}$ and $x\in \mathbb{R}^d$. and $\nabla ...
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A restricted variable increasing convex function property question

There are three variables $x \geq 1, y > 1, z \geq 1$ with $x, y, z \in \mathbb{R}$ and an increasing super-linear convex function $f$ such that $f(x) > x$. If $$x \leq y \cdot z$$ then $$ ...
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59 views

convex hulls of convex sets plus a point

In a real vector space, consider S and T to be disjoint, nonempty, convex subsets of the vector space and let a point x lie outside either set. How would I prove the following: co({x} union S) is ...
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30 views

Why for this equation, convergence is not guaranteed with steepest descent methods?

I uses soft svm recently. I found a good slide from http://www.cs.berkeley.edu/~jordan/courses/281B-spring04/lectures/lec6.pdf. For the soft svm, suppose there are $n$ samples. $f(x_i)=w^Tx_i+b$, ...
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Inequalities of summations lift to inequalities of summations of powers?

Let us assume: an at most countable set of indexes $I$; a set of reals $q_i$ with $i \in I$ such that $0\le q_i \le 1$ and $\sum_{i \in I} q_i \le 1$; a natural $n_i$ for each $i \in I$; a real $x ...
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Conjecture about a property of concave functions

Trying to prove a proposition in my paper, which can potentially use a conjecture about convex (concave) functions. This is likely to be wrong. I appreciate any thoughts on how to prove/disprove this. ...
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How to prove a corollary of this theorem about affine hull?

$\newcommand{\span}{\operatorname{span}}$ $\newcommand{\aff}{\operatorname{aff}}$ Thm: Let $S\subseteq \mathbf{R}^n$. Then for any $m\in \aff S $ (in particular, for any $m\in S$) $\aff S=\left\{ m ...
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Convex polyhedron is union of simplices

Given a convex polyhedron $P$, how can we prove that every point $x \in P$ is in some simplex whose vertices are vertices of $P$? One proof is to inductively build a triangulation of $P$. If $P$ is ...
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Prove that the following two definitions of the convex hull are equivalent.

I was wondering if a topology expert could help me solve this proof, as I have no idea but want to understand these concepts. This is not for homework. Let X be a point set, not necessarily finite, ...
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function defined as an integral involving Bessel functions

i need to analyze a function of the form $$F(x,y) = \int_0^{1} e^{-(1+s)\alpha x}\sinh((1-s)\beta y) I_0(\sqrt{(x^2-y^2)s}) ds $$ Where $I_0$ is the modified Bessel function. $x>y$ always. ...
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Show this function is convex

Show that $f(x)=\frac{(\theta-x)\log_{x}\frac{x-\theta}{1-\theta}+x}{1-x}, ~x\in(\theta,\infty)$ is convex, where $\theta\in(0,1)$. $f(1)=\lim_{x\rightarrow 1}f(x)$. Numerical experiments suggest it ...
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Random convex shapes containing a ball

I'm interested in the properties of randomly generated convex shapes in $n$-dimensional space. Suppose I were to generate $v$ uniformly distributed random points on the $n$-ball of radius $R$. What ...
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Why does the amoeba shrink to its skeleton when we go to infinity?

Let $f\in\mathbb{C}[X_1^{\pm1},\ldots,X_n^{\pm1}]$ a Laurent polynomial. Let $\mathrm{Log}:(\mathbb{C}\setminus\{0\})^n\to\mathbb{R}^n$ defined by ...
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Number of faces of convex hull

If you have $n$ points in $d$-dimensional Euclidean space, the number of faces of the convex hull is potentially exponential I understand. How can this be proved?
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Subdifferential of the sum

Let $C \subset \mathbb R^n$ a nonempty subset. Let us define the indicator function of $C$ $$ I_C(x) = \begin{cases} 0 & x \in C \\ +\infty & x \notin C \end{cases}. $$ Let us consider, in ...
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Analytically solving simple quadratic problem in single variable with boundary constraints

I want to solve the following optimization problem where $x$ is scalar variable. $$ \min_x \dfrac12ax^2 + bx \\ subject\ to:\ l\le x \le u $$ $ a > 0 $ therefore, this is a convex optimization ...
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Terminology for a weaked vector space

Let $S$ be a semigroup on which acts $\mathbb{R}_{\geq0}$. Does this structure has a name? For example $S$ can be the set of convex bodies in $\mathbb{R}^n$ with the Minkowsky sum.
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convexity of a function involving inverse matrix

The function has the following form: $f(a_1,a_2,b_1,b_2) = bA^{-1}c$, where \begin{align*} ...
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If $(\nabla f(x)-\nabla f(y))\cdot(x-y)\geq m(x-y)\cdot(x-y)$, why is $f$ convex?

I was reading on wikipedia that a strongly convex function is also strictly convex. I say that a function $f\colon\mathbb{R}^n\to\mathbb{R}$ is convex if $$ f(\lambda x+(1-\lambda)y)\leq\lambda ...
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A question about a proof for why $\|x\|:=\inf\{\lambda>0\mid\frac{x}{\lambda}\in B\}$ is a norm

I started studying functional analysis, a claim that was thought is the second lecture claims that: Let $X$ be a vector space, $B\subseteq X$ is convex, symmetric around $0$ and s.t ...
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Legendre Transformation of a Lagrangian in Classical Mechanics

I have some questions about the Legendre Transformation of a Lagrangian in Classical Mechanics to the Hamiltonian: We start with a Lagrangian $L(q,\dot{q})=\frac{\langle \dot{q} , \dot{q}\rangle }{2} ...
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Testing for Convexity for a function

Please any one can help figure out if this funcion is concave or convex, any help is greatly appriciated. Any links on how to test fo convexity for such a function is also greatly appriciated. I tried ...
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The convex function proof.

Can anyone help me prove convexity conditions? I know convexity conditions. The second derivative function is greater 0 first order convexity conditions. convex function conditions Because my ...
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106 views

Is $f(x,y) = x^\beta/y$ quasi-convex for positive $x,y$ for any real $\beta \geq 1$?

A multivariate function $f:{\mathbb R}^d \to {\mathbb R}$ is quasi-convex on a convex set $S \subset {\mathbb R}^d$ if $f(\lambda z + (1-\lambda)z') \leq \max\{f(z),f(z')\}$ for all $z,z' \in S$ and ...
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n-simplex in an intersection of n balls

Consider a $n$-simplex, $n \geq 2$ with vertices $x_i,i=1,...,n+1$. For each edge $(i,j)$, consider $n$-ball $B_{ij}$ such that vertices $x_i$ and $x_j$ are antipodal on this ball. Fix a point $x_0$ ...
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How prove this inequality generalized from 1969 IMO problem 6

Let $x_{1},x_{2},y_{1},y_{2},z_{1},z_{2},w_{1},w_{2} $ are all positive numbers, and such $$x_{1}y_{1}z_{1}-w^3_{1}>0,\; \text{ and }\;x_{2}y_{2}z_{2}-w^3_{2}>0.$$ show that ...