# Questions tagged [convexity]

The convexity tag has no usage guidance.

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### 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?

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**1**answer

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_{\...

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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 ...

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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 $...

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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 ...

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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 ...

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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 ...

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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 ...

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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}\...

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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 ...

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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^...

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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 $...

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67 views

### convex approximation for a non convex function

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

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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 $f_k \to f$ ...

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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 ...

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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 $\...

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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 ...

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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) \...

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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(...

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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 $x \in \mathbb{R}^+ \cup\{0\}$
$$
f(x)= \log\left(1+\frac{r}{x+a}\right) + \log\left(1+\frac{r}{2x+a}\right) - 2r \log \left(1+\frac{x}{x+a+r} \right),
$$
...

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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 ...

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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 ...

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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 ...

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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 ...

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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 $K$?

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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 ...

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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\...

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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 ...

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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 ...

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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}{\...

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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 \...

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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( ...

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### 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\...

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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 ...

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### $H_\infty$-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. ...

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### $H_2$-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 ...

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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]{\...

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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)\...

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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,...

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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 \...

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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 $\...

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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 ...

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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 ...

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**1**answer

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
$$\...

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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, ...

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### “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 ...

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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.
...

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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 ...

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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) \...

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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 ...