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Questions tagged [convexity]

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-1
votes
0answers
69 views

What type of convex body would satisfy to its radial function been a spherical polynomial? [on hold]

What type of convex body would satisfy to its radial function been a spherical polynomial?
0
votes
1answer
87 views

Is an ambiguity set with Wasserstein distance of order 1 is convex?

I have a question about the convexity of an Wasserstein ambiguity set. Let $W_1(\mu, \nu)$ be the Wasserstein distance of order 1 between $\mu$ and $\nu$, defined as $$W_1(\mu, \nu) := \min\limits_{\...
4
votes
2answers
366 views

If the closed unit ball of Banach space has at least one extreme point, must the Banach space the be a dual space?

Let $X$ be a Banach space. By Banach-Alaoglu and Krein-Milman Theorems, one can show that if $X$ is a dual space, then $X$ must have at least one extreme point of the closed unit ball. I am ...
2
votes
0answers
88 views

A sufficient condition for subdifferentiability

Corollary 9 in here (page 31) states that a proper convex function $g:Y\rightarrow \mathbb{R}\cup\{\infty\}$ (not necessarily continuous) on a locally convex space $Y$ is subdifferentiable on a point $...
1
vote
0answers
142 views

Is this set in $\mathbb{R}^d$ closed? [closed]

Let $X$ be a convex set (in my case $X$ is infinite dimensional too) and for each $ i \in \{1,2,\dots,d \} $, let $f_{i}: X \rightarrow \mathbb{R} $ be convex functions where $C_{i} := \{ f_{i}(x) : x ...
2
votes
1answer
101 views

Polygon of convex arcs

Convex polygons in the plane $R^2$ arise in linear programming where the constraints are linear. The objective linear function attains its maximum at a vertex of the feasible region(if exists). Assume ...
1
vote
0answers
118 views

The perturbation of a convex function can also be convex?

$ W^{1,\infty}(D)\ni f:D\to\mathbb R, (x,y)\mapsto f(x,y)$, is a strictly increasing on both dimensions (i.e. if $x_1>x_2$ then $f(x_1,y)>f(x_2,y)$), lipschitz continuous function defined on a ...
2
votes
1answer
52 views

Given the following two assumptions, How to conclude $\lim_{k \rightarrow \infty}|\nabla g_r(y^{k})|=0 \ $?

Please refer attached 6-page short paper for details. Let $M(y)=y+2r\nabla g_r(y)$. Given $\ g_r(M(y^k))>g_r(y^k)+r|\nabla g_r(y^k)|^2, \forall k. \ $ (equation 36 in the paper) and $ \ \lim_{k ...
1
vote
0answers
47 views

Convexity on large scales

Has the following concept ever been studied/have a standard name? Let $f:\mathbb{R}\to \mathbb{R}$ be continuous. We say $f$ is (mid-point) convex on large scales provided $$f\left( \frac{x+y}{2}\...
1
vote
0answers
29 views

Weak convexity in graphs

I asked the following question in MathStackExchange, but I did not get any answer, and I think that this might be the appropriate venue for the question. As we know, a finite undirected graph ...
6
votes
0answers
90 views

Geometric mean of three or more positive definite matrices

The geometric mean of two positive definite (Hermitian) matrices of same size is defined by $$A\natural B := A^{1/2}(A^{-1/2}BA^{-1/2})^{1/2}A^{1/2},$$equivalently, $$A\natural B =(BA^{-1})^{1/2}A=A(A^...
4
votes
1answer
185 views

An inequality of T. Carleman

I'm looking for the name and some references for the proof of the inequality below. I founded that is due to T. Carleman but no reference was given. Let $f(z)$ be an analytic function on a subdomain $...
2
votes
0answers
67 views

convex approximation for a non convex function

Consider the function $f\left( {{x_1},...,{x_M},{y_1},...,{y_N}} \right) = \left( {\sum\limits_{j = 1}^M {{\alpha _j}{x_j}} } \right)\left( {{e^{ - \sum\limits_{i = 1}^N {{\beta _i}{y_i}} }}} \right)$...
5
votes
0answers
74 views

Positive splitting of Sobolev convergence

Let $f,g,h \in H^1(\mathbb{R}^n)$ be non-negative Sobolev functions wuch that $f^2 = g^2 + h^2$. Let also $\{f_k\} \subseteq H^1(\mathbb{R}^n)$ be non-negative Sobolev functions such that $f_k \to f$ ...
16
votes
1answer
505 views

Extreme points of convex compact sets

Preparing to a lecture on Krein--Milman theorem I read in W. Rudin's Functional analysis textbook (1973) that it is unknown whether any convex compact set in any topological vector space has an ...
2
votes
1answer
90 views

Quotient with positive second derivative in the limit?

I am studying the quotient of $$f(\varepsilon) = \sum_{i=1}^{\infty} \frac{i^2}{2^{\varepsilon i^2}}$$ and $$g(\varepsilon) = \sum_{i=1}^{\infty} \frac{1}{2^{\varepsilon i^2}}$$ for some $\...
0
votes
3answers
95 views

On the properness of the graph of a convex function

Let $f : \Omega \subseteq \mathbb{R}^n \to \mathbb{R}$ be a smooth and convex function. Let us assume that $\Gamma_f = \mathrm{graph}(f) $ is a complete hypersurface of $\mathbb{R}^{n+1}$. Then I know ...
3
votes
1answer
190 views

concavity of $\log [ (1+\frac{x_0}{1+x_0+x_1/2+x_2/4}) (1+\frac{x_1}{1+x_0/4+x_1+x_2/2}) (1+\frac{x_2}{1+x_0/2+x_1/4+x_2}) ]$

Let $f:~ [0,1]^3 \rightarrow \mathbb{R}$ be $$ f(x_0,x_1,x_2)= \log \left[ \left(1+\frac{x_0}{1+x_0+\frac{x_1}{2}+\frac{x_2}{4}}\right) \left(1+\frac{x_1}{1+x_1+\frac{x_2}{2}+\frac{x_0}{4}}\right) \...
2
votes
0answers
52 views

Star-convex curve and Fourier series

Let x(t) be a periodic function on [0, 2$\pi$]. I am interested in finding criteria in terms of the Fourier coefficients of x(t), such that the parametric curve $\left\{ x\left( t\right) ,\dot{x}\left(...
-6
votes
1answer
95 views

Behavior of $f(x)= \log\left(1+\frac{r}{x+a}\right) + \log\left(1+\frac{r}{2x+a}\right) - 2r \log \left(1+\frac{x}{x+a+r} \right)$ [closed]

Consider the following function defined on $x \in \mathbb{R}^+ \cup\{0\}$ $$ f(x)= \log\left(1+\frac{r}{x+a}\right) + \log\left(1+\frac{r}{2x+a}\right) - 2r \log \left(1+\frac{x}{x+a+r} \right), $$ ...
4
votes
3answers
143 views

Quasi-concavity of $f(x)=\frac{1}{x+1} \int_0^x \log \left(1+\frac{1}{x+1+t} \right)~dt$

I want to prove that function \begin{equation} f(x)=\frac{1}{x+1} \int\limits_0^x \log \left(1+\frac{1}{x+1+t} \right)~dt \end{equation} is quasi-concave. One approach is to obtain the closed form ...
0
votes
0answers
46 views

Choquet Theorem for the cone of non-negative operators

Let $\mathcal B_+$ be the convex cone of bounded non-negative self-adjoint operators on $L^2(\mathbb R)$. With the perspective of applying Choquet's Theorem, I would like to know if the extreme points ...
5
votes
1answer
105 views

Cutting a convex body into two congruent pieces

This question is related to How to make a sandwich from just one piece of bread?, asked on Feb 23 '17 by erz, and it goes as follows: Question. If a convex closed and bounded region $C$ in the ...
2
votes
1answer
110 views

Quasi-concavity of $f(x)=(1-\frac{x}{1000})\log_2(1+2^x)$ on $[0~1000]$

I want to prove that function $f:[0~1000]\rightarrow R$, $$f(x)=(1-\frac{x}{1000})\log_2(1+2^x)$$ is quasi-concave. Any idea how to do the proof? I already tried to prove that any super-level set is ...
3
votes
1answer
124 views

Convexity of the matrix mapping $X^{-2}$

Let $X$ be a positive semidefinite matrix. Is the mapping $X\to X^{-2}$ convex? Update: or is $Tr[X^{-2} K]$ convex for PSD $X$ and $K$?
1
vote
0answers
62 views

Convexity of the variance of a function depending on random variables

here is my question: I have a function $f(x,\epsilon_1, \dots, \epsilon_n)$ that depends on a decision $x$ I make and a certain amount of random variables $\epsilon_i$. I define the two following ...
3
votes
1answer
183 views

A measure of noncompactness by a convex function

Let $E, \left \| \right \|$ be a Banach space, $\mathfrak{M}_E$ indicate a family of all nonempty bounded subset of $E$, $\mathfrak{N}_E$ the familly of all relatively compact sets, and $Ker \mu=\{X\...
3
votes
2answers
196 views

Extreme points of a convex set

Let $S$ denote the set of all complex non-negative definite matrices with all diagonal elements being less that or equal to one. Can we show that any matrix which belongs to the set of all non-zero ...
5
votes
1answer
240 views

An inequality involving a sum of power terms

I am currently working in a problem in Information Theory and I came across a difficult inequality. After many attemps, I simplified the inequality, which now looks at follows. Consider a positive ...
1
vote
0answers
42 views

Convex solutions of linear hyperbolic PDEs in a planar domain

Consider a linear homogeneous 2nd-order PDE in a convex planar domain $\Omega$ : $$a(x,y)\frac{\partial^2u}{\partial x^2}+2b(x,y)\frac{\partial^2u}{\partial x\partial y}+c(x,y)\frac{\partial^2u}{\...
1
vote
0answers
54 views

Condition on $f := (I-\delta P)^{-1}g$ to ensure convexity of $f$?

Consider the following extension of the notion of convexity of continuous functions to functions defined over $\mathcal{I} = \{0,\ldots,n \}$ — that is, to vectors: $f$ is convex if for any $\alpha \...
11
votes
2answers
326 views

Convex hull of the Stiefel manifold with non-negativity constraints

Consider the Stiefel manifold $$\mathrm{St}(n,k) :=\{X \in \mathbb{R}^{n\times k} : X^TX = I_k\},$$ where $I_k$ is the $k$-dimensional identity matrix. It is well known that $$\mathrm{conv} \left( ...
0
votes
0answers
14 views

Structural properties (higher order monotonicity) preserved by linear differential operators

Consider the following linear differential equation: $(\rho+x)\ f_i(x,y)+k(x-b)\ \partial_x\ f_i(x,y)=x\ f_{i-1}(x,y),\quad x\geq x_0>b>0$ where $i\geq1$, $f_0(x,y)=0$ for any $(x,y)\in\...
0
votes
0answers
115 views

Convexity of the Frobenius norm of the product of matrices

I have a question similar to Convexity of the Frobenius norm of the product of two matrices. I am not able to comment on that question as I don't have enough reputation, and that is why I have asked a ...
0
votes
0answers
47 views

$H_\infty$-norm of a time delay system

I have a linear dynamical stable delayed system as follows, $\dot{x}=Ax(t-\tau)+Bu$, where $A_{n \times n}$ is a stable matrix with all its eigenvalues located on the open left half of complex plane. ...
0
votes
0answers
42 views

$H_2$-norm of a time delay system

I have a linear dynamical stable delayed system as follows, $\dot{x}=Ax(t−τ)+Bu$, where $A_{n \times n}$ is a stable matrix with all its eigenvalues located on the open left half of complex plane. I ...
1
vote
0answers
79 views

Are Outer Products of Sub-Gaussian Vectors Sub-Exponential?

$\newcommand\xx{\mathbf{x}}\newcommand\yy{\mathbf{y}}\newcommand\A{\mathbf{A}}\newcommand\aalpha{\boldsymbol{\alpha}}\newcommand\bbeta{\boldsymbol{\beta}}\newcommand\E{\mathbb{E}}\newcommand\inner[1]{\...
0
votes
0answers
55 views

continuous map from $\mathbb R$ to $\mathbb R^2$ that send any convex on a convex [duplicate]

Let $f$ be a continuous fonction from $\mathbb R$ to $\mathbb R^2$, such that for any $a<b\in \mathbb R,\,\, f([a,b])$ is convex. Is there a line $D\subset \mathbb R^2$ such that $f(\mathbb R)\...
5
votes
0answers
100 views

Dimensions of faces of convex hull of convex bodies

Let $K_1,\ldots,K_m\subset\mathbb{R}^n$ with $m\geq n$ some convex bodies (i.e. compact with nonempty interior). I am interested in sufficient criteria for the convex hull $K=\textrm{conv}(K_1,\ldots,...
2
votes
1answer
60 views

Common boundary point of convex bodies

Let $K_1, \ldots, K_n \subset \mathbb{R}^n$ be some convex bodies (i.e., compact with nonempty interior) such that for each subset $I \subset \{1,\ldots,n\}$ the set $$\left( \bigcap_{i \in I} K_i \...
8
votes
1answer
238 views

Almost convex combinations in $\mathbb R^n$

Working on some problems in the $C_p$-theory I discovered the following simple but amazing Fact. For any subset $A\subset \mathbb R^n$, non-zero vector $a\in \bar A\subset\mathbb R^n$ and $\...
3
votes
1answer
43 views

Separation of Infinite-Dimensional Salient Convex Cones

Let X be the set of all summable sequences of reals endowed with the $l^1$ norm. That is, two elements of x are $a=(a_1,a_2,....)$ and $b=(b_1,b_2,...)$ and $d(a,b) = \sum_n |a_n-b_n|$. In this set ...
1
vote
1answer
78 views

Nonlinear low-rank approximation - corrected

I would like to state that this is related to a past question of mine which contained errors and now appears in the corrected form, with the erroneous one deleted and closed. In my research of linear ...
1
vote
1answer
82 views

Inequality satisfied for $t=1/2$ and Measurability implies Log-Concavity

Say $f:\mathbb{R}\to(0,\infty)$ is measurable, and that $$\forall a,b\in\mathbb{R}~~a<b\implies\log f(a)+\log f(b)\leq2\log f(\frac{a+b}{2}).$$ Why must $f$ be log-concave? (That is, why must $$\...
3
votes
2answers
214 views

Mixtures of log-convex functions are log-convex: a reference

A referee of a submitted paper requested details on the statement that $\int_0^a e^{-tx^2}\,dx$ is log-convex in real $t$, for each $a>0$. While there are a number of ways to prove this statement, ...
4
votes
0answers
61 views

“Singularly convex” cones of matrices

The ambient space if ${\bf M}_n({\mathbb R})$. Let us begin with facts. 1- The cone of positive semi-definite symmetric matrices is convex. 2- It is a little subtler that the cone $K^+$ of matrices ...
0
votes
0answers
70 views

Formalize intuitive observations about plane convex sets

I need to prove very specific assertions concerning convex sets of $\mathbb{R}^2.$ However, despite the very intuitive aspect of these propositions, the formal aspect is pretty hardcore for me. ...
8
votes
2answers
264 views

Faces of the intersection of convex sets

Let $V$ be a normed real vector space and let $K_1, K_2\subseteq V$ be closed convex subsets such that the intersection $K_1\cap K_2$ is non-empty. Assume that $F_1$ is a face of $K_1$ and $F_2$ is a ...
2
votes
0answers
135 views

Orthonormal Basis for Convex Functions

Are there any orthonormal bases for strictly convex functions $f: \mathbb{R}^n\ni x \mapsto \mathbb{R},\ x\ne y\implies f\left(\alpha x+\left(1-\alpha\right)y\right) \lt \alpha f(x)+(1-\alpha)f(y) \...
8
votes
0answers
182 views

Plank invariant measures on convex bodies

Let $K\subset R^2$ be a convex body, i.e., a compact convex set with interior points. A plank $P$ is the region between a pair of parallel lines in $R^2$. Let us say that $P$ intersects $K$ properly ...