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2
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0answers
51 views

A sufficient condition for subdifferentiability

Corollary 9 in here (page 31) states that a proper convex function $g:Y\rightarrow \mathbb{R}\cup\{\infty\}$ on a locally convex space $Y$ is subdifferentiable on a point $y$ in the quasi-relative ...
3
votes
0answers
58 views

Boundedness of Legendre transform domain for convex functions

Let $\Omega$ be a bounded convex domain in $\mathbb{R}^n$ and $u$ be a continuous convex function on the closure $\overline{\Omega}$. Given a boundary point $p\in\partial\Omega$, $u$ is said to have ...
1
vote
0answers
140 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
99 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
113 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
51 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
42 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
87 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
172 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
61 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
70 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 $f_k \to f$ ...
14
votes
1answer
436 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
88 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
87 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
186 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
50 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
94 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
141 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 ...
4
votes
1answer
102 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
123 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
55 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 ...
2
votes
1answer
161 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
187 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
221 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
40 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
318 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
101 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
46 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
41 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
70 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
50 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)\...
0
votes
0answers
61 views

Topological Vectorspaces which are both Mackey and weakly topologized

1) Given two topologies $\mathcal{T}_1, \mathcal{T}_2$ on $X$, we say $\mathcal{T}_1$ is weaker than $\mathcal{T}_2$ if $\mathcal{T}_1\subseteq\mathcal{T}_2$. Let $X$ be a set and $(Y_i)_{i \in I}$ a ...
5
votes
0answers
99 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
235 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
75 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
194 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
69 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
251 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
127 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
178 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 ...
32
votes
0answers
909 views

Converse of the Archimedean property of the sphere

In his remarkable book On the Sphere and Cylinder, where he came tantalizingly close to discovering calculus, Archimedes showed that the area of the portion of the sphere contained between a pair of ...