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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Convexity of inverted sum of positive definite matrices

I am currently working with a class of functions, where every function looks like $f(x)=v^T(xA+(1-x)B)^{-1}v,$ where $v\in\mathbb{R}^n$ is an arbitrary vector, $A,B \in \mathbb{R}^{n\times n}$ are ...
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469 views

Differentiability of Moreau-Yosida approximation.

I want to show that if $X$ is a reflexive Banach space with norm of class $\mathcal{C}^1$ and $f\colon X\to\mathbb{R}\cup \{+\infty\}$ is convex and lower semicontinuous, then $f_{\lambda}$ is ...
4
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1answer
839 views

Proving Helly's theorem

The problem is to prove Helly's theorem for the case, when the convex bodies are polytopes, by using linear programming duality. Helly's theorem Let $B_{1},...,B_{m}$ be a collection of convex ...
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1answer
114 views

Describing a set of normals

In a finite dimensional (think Euclidean) ambient space, let $S$ a compact, convex set and $x$ not in $S$. The two sets can be (weakly) separated, i.e. there exists a vector (normal) that defines a ...
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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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1answer
48 views

When the closure of a convex set contains a ball

Suppose $C$ is a convex set in $\mathbb{R}^n$ whose closure contains the open ball $B(x,r)$. Is it true that $C$ contains $B(x,r)$? Motivation: I am asking this because something like this seems to ...
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155 views

Product / GM of numbers, with fixed mean, increase as numbers get closer to mean.

I am trying to prove a statement which goes like this. Let $a_i$ and $b_i$ be positive real numbers where $i = 1,2,3,\ldots,n$; where $n$ is a positive integer greater than or equal to $2$, such ...
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210 views

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 ...
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116 views

Convexity of Exponential Composite Function

$f:\mathbb{R}_{+}^M\rightarrow\mathbb{R}_+$ is a convex analytic function. For $\mathbf{x}\in\mathbb{R}_{+}^M$ and $y\in\mathbb{R}_{+}$, consider the function ...
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204 views

strict convexity with a measure theoretic property

Suppose $(x_n)$ is a positive sequence of reals converging to $x$. Furthermore we have a measure space $(E,\mathcal{E},\mu)$ given, with finite measure $\mu$. There are a measurables nonnegative and ...
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$\int_0^1 u(t)\phi''(t)dt \geq 0,\ \forall \phi\in C_0^1((0,1)), \ \phi\geq 0$. Is $u$ convex?

Suppose that $u\in C([0,1])\cap C^1((0,1))$ satisfies for all $\phi\in C_0^2((0,1))$, $\phi\geq 0$ $$\int_0^1 u(t)\phi''(t)dt \geq 0$$ Can we conclude that $u$ is convex? Note: $C_0^2((0,1))$ is ...
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189 views

Convex Alternatives to the Gamma Function?

The Bohr-Mollerup Theorem states that the gamma function is the unique function $f: (0, \infty )\rightarrow \mathbb{R}$ satisfying $f(1)=1,$ $f(x+1) = x f(x),$ and the condition that $\log f$ is ...
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1answer
242 views

Effect the zero vector has on the dimension of affine hulls and linear hulls

I am currently working through "An Introduction to Convex Polytopes" by Arne Brondsted and there is a question in the exercises that I would like a hint, or a nudge in the right direction, please no ...
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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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85 views

Convexity of $S=\left\{ \frac{\underline{x}^HA\underline{x}}{\underline{x}^H\underline{x}} , \underline{x} \in \mathbb C ^{n}\right\}$

i've to prove that the following set is convex: Let $A \in \mathbb{C}^{n \times n}$ $$S=\left\{ \frac{\underline{x}^HA\underline{x}}{\underline{x}^H\underline{x}} , \underline{x} \in \mathbb C ...
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Properties of the square norm in Banach spaces

Let $X$ be a Banach space with its dual $X^*$. Consider the mapping $f: X\rightarrow \mathbb{R}$ given by $$ f(x)=\frac{1}{2}\|x\|^2. $$ We have know that when $X$ is a real Hilbert space ($X=X^*$) ...
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95 views

Number of supporting hyperplanes

I know that, for any convex set $S$, there is at least one supporting hyperplane at every point in $B$, the boundary of $S$. Also, there can be more than one supporting hyperplane at the same point in ...
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107 views

Averages of a Function

Let $(X,M,\mu)$ a measure space and $f:X \rightarrow \mathbb{C}$ a function in $L^{\infty}(\mu)$. Define $A_f$ as the set of all averages \begin{equation} \frac{1}{\mu(E)} \int_{E} f d \mu ...
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104 views

On a version of gradient descent

I am trying to read this paper and have gotten stuck. The author considers the problem of minimizing a convex function whose gradient has coordinate-wise Lipschitz constant $M$ (meaning that for all ...
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101 views

Convexity of polylogarithms

I want to prove the following proposition: The function $w\to (-Li_{5/2}(-e^w))^{2/5}$ is convex on $\mathbb R$. And, as I think, the same is true for the function $w\to (-Li_{p}(-e^w))^{1/p}$ for ...
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Space of solutions to $f(x+y) = f(x) + f(y)$ when $f$ is convex

Let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ be a nonlinear convex function, and let $x\in\mathbb{R}^n$ be an arbitrary vector. Define the set $\Omega_f(x)$ as $\Omega_f(x) \triangleq ...
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79 views

Is there a notion of 'pre-convex' sets?

A set $A$ in $\mathbb{R}^n$ is convex if for any two points $p,q \in A$ and real $\lambda \in [0,1]$, the point $\lambda p + (1-\lambda)q$ is also in $A$. There are many beautiful theorems about ...
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Mappings preserving convex polyhedra

It is known that linear mappings between euclidean spaces map convex polyhedra to convex polyhedra. Can you give a characterization of the class of mappings that preserve convex polyhedra?
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Cyclically monotone sets on four points

A subset of $\mathcal{X} \times \mathcal{Y} \subset \mathbb{R}^d \times \mathbb{R}^d$ is cyclically monotone if $\sum_{i=1}^n \langle x_i,y_i\rangle \ge \sum_{i=1}^n \langle x_i,y_{i+1}\rangle$, where ...
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147 views

Convexity of a Given Function

Is the following function convex or concave? $$f(\mathbf{x}) = \frac{1}{N}\sum_{i=1}^N\sqrt{1+\frac{x_i^2}{\left(\frac{1}{N}\sum_{i=1}^{N}x_i\right)^2}},$$ $\mathbf{x} = [x_1,x_2,...,x_N]^T, x_i \ge ...
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263 views

Convexity of cubic vector function?

First question in these forums so go easy on me. I have a function $f_i(x):\mathbb{R}^N\to\mathbb{R}$ which is defined by $$f_i(x) = \frac{(x^TAe_i)x^TAx}{(x^TA+b^T){\bf 1}}$$ where ...
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Find optimal measure

Let $\Omega$ be a convex compact set in $\mathbb{R}^n$, $f\colon \Omega \to \mathbb{R}$ be a convex function. Consider an optimization problem $$ \int\limits_{\Omega}f(x)\,\mu(dx) \to ...
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Prove one set is a convex hull of another set

Define two sets: $A = \{x \in \{0,1\}^n : \lVert x \rVert_1 \leq k\}$ is a finite set of binary vectors; $B = \{x \in [0,1]^n : \lVert x \rVert_1 \leq k\}$ is an infinite set of real-valued vectors, ...
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85 views

A question about pyramids (polytopes)

Let's use the following definition of a face: A nonempty convex subset $F$ of a convex set $C$ is called a face of $C$ if $\alpha x + (1-\alpha) y \in F$ with $x, y \in C$ and $0 < \alpha < 1$ ...
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Show by example that the Minkowski sum of two sets $X+Y$ may be convex even if neither $X$ nor $Y$ are convex

There were two parts to this question. I proved that the Minkowski sum of two sets $X+Y$ is convex whenever $X$ and $Y$ are convex, but how do I prove this second part? "Show by example that the ...
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323 views

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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1k views

How to prove convexity?

Let us consider the function $$I(p):= \frac {\Gamma(2-p)\Gamma(3p)}{(p\Gamma(p))^2} $$ on the interval $(0,1),$ where $\Gamma(x)$ denotes the gamma function. How to prove its convexity there?
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Show that $f(x)=||x||^p, p\ge 1$ is convex function on $\mathbb{R}^n$

Show that $f(x)=||x||^p, p\ge 1$ is convex function on $\mathbb{R}^n$. I have tried to use Holder's inequality, but I still cannot solve this problem. Could you help me with this problem? Thank you ...
3
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208 views

How can I prove that a function $f(x,y)= \frac{x^2}{y}$ is convex for $y \gt 0$?

How can I prove that a function $f(x,y)= \frac{x^2}{y}$ is convex for $ y \gt 0$? I take the Hessian matrix of $\displaystyle \frac{x^2}{y}$, and I got: $$H = \displaystyle\pmatrix{\frac{2}{y} & ...
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1answer
105 views

Equations for interior of Platonic solids

It is well-known that for Platonic solids: The interior of cube a.k.a. hexahedron can be described with inequality $\max\{|x|,|y|,|z|\}<a$. The interior of octahedron is $|x|+|y|+|z|<a$. But ...
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186 views

Sufficient condition for a function to be affine

If for a function $f:\mathbb{R}\to\mathbb{R}$, I can prove for any real $x,y$, that $f(\frac{x+y}{2})=\frac{f(x)}{2}+\frac{f(y)}{2}$, can I say for sure that it is affine, as in of the form ...
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Using the definition of a concave function prove that $f(x)=4-x^2$ is concave (do not use derivative).

Let $D=[-2,2]$ and $f:D\rightarrow \mathbb{R}$ be $f(x)=4-x^2$. Sketch this function.Using the definition of a concave function prove that it is concave (do not use derivative). Attempt: ...
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Convexity of intersection

I have been asked to prove that, given a convex set $C$, its intersection with a line is also convex. From convexity definition, I have that $\forall x_1,x_2\in C, \alpha x_1+\beta x_2 \in C$ with ...
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Euclidean Metric and Convexity

Question: Consider the Euclidean metric space $(\mathbb{R}^n , \Vert\cdot \Vert)$. Let $X\subset \mathbb{R}^n$ and $f\colon X \to \mathbb{R}$. $X$ is said to be a convex set if for every $x,y \in X$ ...
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Show that $(1+x)(1+y)(1+z)\ge 8(1-x)(1-y)(1-z)$.

If $x>0,y>0,z>0$ and $x+y+z=1$, prove that $$(1+x)(1+y)(1+z)\ge 8(1-x)(1-y)(1-z)$$ Trial: Here $$(1+x)(1+y)(1+z)\ge 8(1-x)(1-y)(1-z) \\ \implies (1+x)(1+y)(1+z)\ge 8(y+z)(x+z)(x+y)$$ I ...
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2answers
160 views

Property of convex functions

I am trying to show that if a function $f$ defined in $\mathbb R^n$ is differentiable and convex then $f(y)-f(x)\ge \nabla f(x)(y-x).$ for each $x,y\in\mathbb R^n$ Using differentiability of $f$ I ...
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2answers
101 views

Is $g(x)=\log x$ convex function?

The graph of convex function is : In a book it is written that $g(x)=\log x$ is strictly convex function. So i searched for graph of $g(x)=\log x$ and found that Though it has been said that ...
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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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3answers
166 views

Distributed Convex Optimization Algorithm

Consider the convex optimization problem $$ \min_{x_1, \cdots, x_N, y} \sum_{i=1}^{N} f_i(x_i,y) $$ $$ \text{subject to: } x_i \in X_i \ \ \forall i, \ \ y \in Y, \ \ y = \sum_{i=1}^{N} x_i $$ ...
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291 views

Convex Sets in Functional Analysis?

Why did Bourbaki choose to study convex sets, convex functions and locally convex sets as part of the theory of topological vector spaces, and what is so important about these concepts? I'd like to ...
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447 views

A standard quadratic minimization problem

Consider the "Complex" Quadratic minimization problem \begin{align} \min_{\mathbb{x}\in \mathbb{C}^{N \times 1}}~\mathbf{{x}}^H\mathbf{Q}\mathbf{x}-2~\Re{(\mathbf{x}^H\mathbf{b})}+1 \end{align} ...
3
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1answer
389 views

Two questions regarding the ergodic decomposition theorem

In Walters' An Introduction to Ergodic Theory, page 153, Remark (2), he writes If $E(X,T)$ denotes the set of extreme points of $M(X,T)$ then for each $\mu \in M(X,T)$ there is a unique measure ...
3
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2answers
317 views

Gradient-descent algorithm always converges to the closest local optima?

Assume $f(\vec x)$, which is Lipschitz continuous, has two local optima $\vec x_1^*$ and $\vec x_2^*$( $\vec x_1^*$ is the global minimum). We start the gradient-descent algorithm from $\vec x_0$ and ...
3
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2answers
81 views

Conjugate convex functions property

$E$ is normed vector space.Let $f\in E^*$ in a bounded linear functional from $E$ to $C$ and fix $x\in E$. We have $$\forall y\in E;\ \ \ \ f(y-x)\leq \frac{1}{2}\|y\|^2-\frac{1}{2}\|x\|^2$$ And I ...
3
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1answer
57 views

How to prove the compactness of the set of Hermitian positive semidefinite matrices

I am dealing with convex optimization problems. There are some useful theories for optimization problems where real-valued vector parameter, e.g., $x \in \mathbb{R}^n$, is considered. I manage to ...