Questions tagged [convexity]
For questions involving the concept of convexity
625
questions
2
votes
1
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Closest points of curves on convex surfaces
Let $\Sigma$ be a convex surface with strictly positive curvature in $\mathbb{R}^3$ and $\Gamma \subset \Sigma$ be a closed simple space curve with parameterization $\gamma : \mathbb{S}^1 \rightarrow \...
0
votes
1
answer
107
views
Subadditive function with special growth
Related to one of my previous question (for which I have received an answer, thanks) I have the following new one. Maybe I am describing the empty set but not being a specialist at all of the domain I ...
6
votes
1
answer
194
views
How to calculate the volume of a section of a convex body?
The following is essentially a partial case for my previous question.
Let $B\subset\mathbb{R}^m$ be the unit ball with respect to a concrete norm on $\mathbb{R}^m$, say $l^p$-norm, $p\in (1,\infty)$....
4
votes
1
answer
1k
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Simple proof of Prékopa's Theorem: log-concavity is preserved by marginalization
The following result is well-known:
Suppose that $H(x,y)$ is a log-concave distribution for $(x,y) \in \mathbb R^{m \times n}$ so that by definition we have
$$H \left( (1 - \lambda)(x_1,y_1) + \...
1
vote
0
answers
33
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Modelling exact unions of polytopes in homogeneous case?
We can model disjunctions (note I am not looking for convex hull) of $t$ unbounded convex polyhedra given by $A^{(1)}x^{(1)}\leq b^{(1)}$,$\dots$,$A^{(t)}x^{(t)}\leq b^{(t)}$ exactly with a mixed ...
1
vote
0
answers
104
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John's ellipsoid of a polytope
Suppose that $X$ is $\mathbb R^n$ with some polyhedral norm, that is, the unit ball of $X$ is an $n$-dimensional polytope. Assume that the John ellipsoid of $X$ is an Euclidean ball that touches every ...
2
votes
0
answers
89
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Neumann problem on a convex domain
Let $\Omega$ be a convex open bounded subset of $\mathbb R^n$ and let $u$ be the solution of
$$
\begin{cases}
∆ u=1\quad\text{in $\Omega$,}
\\
\frac{\partial u}{\partial \nu}=\frac{\vert \Omega\vert_n}...
4
votes
1
answer
146
views
Mapping inclusion theorem for the numerical range
We denote the numerical range of a complex square matrix $A \in \mathbb{C}^{n\times n}$ by $W(A)$.
Let $A \in \mathbb{C}^{n\times n}$ and let $f: \mathbb{C} \to \mathbb{C}$ be, say, an entire ...
4
votes
1
answer
296
views
On convex functions which are non constant on every segment
I have been studying for the last few weeks the paper Dirichlet problem for demi-coercive functionals by Anzellotti, Buttazzo and Dal Maso. In this work the authors introduced and studied the ...
3
votes
1
answer
439
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Is there exists (strictly) convex function on hemisphere?
Given $\mathbb{S}^n_+:=\{x\in \mathbb{R}^{n+1}: |x|=1,x_{n+1}>0\}$ be the open domain in $\mathbb{S}^n$, or be viewed as the geodesic ball centered at the pole with radius $\frac{\pi}{2}$ in $\...
0
votes
0
answers
54
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linear functions/hyperplanes vs. convex functions/convex sets in Hilbert space
The simplest Hahn-Banach extension theorem in Hilbert space $X$ avoids the use of the axiom of choice by virtue of the Riesz representation theorem. But what about the version of the theorem where the ...
0
votes
1
answer
107
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Inequality involving product-of-minus vs minus-of-product for positive integers
I'm encountering this inequality for dimensionality reduction problem. The simplified form looks as follows:
Consider positive integers $a_1$, $a_2$, $b_1$ and $b_2$ where $a_1>b_1$ and $a_2>...
1
vote
1
answer
442
views
Is there a way to turn a non convex set to a convex one? [closed]
Perhaps the question is rather vague, but if we are given a non-convex set S, can we construct some invertible mapping f so that f(x) becomes a convex set?
In my problem , I am given a set of matrix ...
0
votes
0
answers
69
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Number of local minima of a particular non-convex composite function
I am interested in proving that the following equation on the interval $x \in [c,1]$ is minimized either at the endpoints or where $x=\sqrt{c}$:
$f(x)=\frac{-1}{\log(1-x)}+\frac{-1}{\log(1-\frac{c}{x}...
0
votes
1
answer
89
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Existence of a certain type of function
Trying to find functions with the given property:
Given $M>0, K$ compact in $\mathbf{R^n}$,$f:U\rightarrow\mathbf{R}$ a $C^2$ function, where $U$ open in $\mathbf{R^n}$ and $K\subset U$such that $...
8
votes
2
answers
485
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How can we show that if $f$ is convex, then $\liminf_{|x|\to\infty}\frac{x\cdot\nabla f(x)}{|x|}>0$?
Let $d\in\mathbb N$ and $f:\mathbb R^d\to\mathbb R$ be convex with $$\int e^{-f(x)}\:{\rm d}x<\infty\tag1.$$ How can we show that $$\liminf_{|x|\to\infty}\frac{x\cdot\nabla f(x)}{|x|}>0?$$ $f$ ...
10
votes
0
answers
143
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A geometrical problem in terms of a convex function
I wish to know whether the following problem has ever been investigated.
Let $D$ be a convex domain in ${\mathbb R}^d$, with smooth boundary $\partial D$. Let $\vec V:\partial D\rightarrow{\mathbb R}^...
4
votes
1
answer
2k
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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
$$W_1(\mu, \nu) := \min\limits_{\...
4
votes
2
answers
1k
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If the closed unit ball of Banach space has at least one extreme point, must the Banach space the be a dual space?
Let $X$ be a Banach space.
By Banach-Alaoglu and Krein-Milman Theorems, one can show that if $X$ is a dual space, then $X$ must have at least one extreme point of the closed unit ball.
I am ...
1
vote
0
answers
148
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Is this set in $\mathbb{R}^d$ closed? [closed]
Let $X$ be a convex set (in my case $X$ is infinite dimensional too) and for each $ i \in \{1,2,\dots,d \} $, let $f_{i}: X \rightarrow \mathbb{R} $ be convex functions where $C_{i} := \{ f_{i}(x) : x ...
3
votes
1
answer
245
views
Polygon of convex arcs
Convex polygons in the plane $R^2$ arise in linear programming where the constraints are linear. The objective linear function attains its maximum at a vertex of the feasible region(if exists).
Assume ...
1
vote
0
answers
375
views
The perturbation of a convex function can also be convex?
$ W^{1,\infty}(D)\ni f:D\to\mathbb R, (x,y)\mapsto f(x,y)$, is a strictly increasing on both dimensions (i.e. if $x_1>x_2$ then $f(x_1,y)>f(x_2,y)$), lipschitz continuous function defined on a ...
2
votes
1
answer
63
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Given the following two assumptions, How to conclude $\lim_{k \rightarrow \infty}|\nabla g_r(y^{k})|=0 \ $?
Please refer attached 6-page short paper for details.
Let $M(y)=y+2r\nabla g_r(y)$.
Given $\ g_r(M(y^k))>g_r(y^k)+r|\nabla g_r(y^k)|^2, \forall k. \ $ (equation 36 in the paper)
and $ \ \lim_{k ...
2
votes
0
answers
60
views
Convexity on large scales
Has the following concept ever been studied/have a standard name?
Let $f:\mathbb{R}\to \mathbb{R}$ be continuous. We say $f$ is (mid-point) convex on large scales provided
$$f\left( \frac{x+y}{2}\...
2
votes
0
answers
56
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 ...
8
votes
0
answers
184
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,
$$A\natural B =(BA^{-1})^{1/2}A=A(A^...
4
votes
1
answer
265
views
An inequality of T. Carleman
I'm looking for the name and some references for the proof of the inequality below. I founded that is due to T. Carleman but no reference was given.
Let $f(z)$ be an analytic function on a subdomain $...
3
votes
0
answers
114
views
convex approximation for a non convex function
Consider the function
$f\left( {{x_1},...,{x_M},{y_1},...,{y_N}} \right) = \left( {\sum\limits_{j = 1}^M {{\alpha _j}{x_j}} } \right)\left( {{e^{ - \sum\limits_{i = 1}^N {{\beta _i}{y_i}} }}} \right)$...
6
votes
0
answers
112
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Positive splitting of Sobolev convergence
Let $f,g,h \in H^1(\mathbb{R}^n)$ be non-negative Sobolev functions such that $f^2 = g^2 + h^2$. Let also $\{f_k\} \subseteq H^1(\mathbb{R}^n)$ be non-negative Sobolev functions such that $f_k \to f$ ...
17
votes
1
answer
832
views
Extreme points of convex compact sets
Preparing to a lecture on Krein--Milman theorem I read in W. Rudin's Functional analysis textbook (1973) that it is unknown whether any convex compact set in any topological vector space has an ...
2
votes
1
answer
94
views
Quotient with positive second derivative in the limit?
I am studying the quotient of
$$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}}$$
for some $\...
0
votes
3
answers
110
views
On the properness of the graph of a convex function
Let $f : \Omega \subseteq \mathbb{R}^n \to \mathbb{R}$ be a smooth and convex function. Let us assume that $\Gamma_f = \mathrm{graph}(f) $ is a complete hypersurface of $\mathbb{R}^{n+1}$. Then I know ...
4
votes
1
answer
282
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concavity of $\log [ (1+\frac{x_0}{1+x_0+x_1/2+x_2/4}) (1+\frac{x_1}{1+x_0/4+x_1+x_2/2}) (1+\frac{x_2}{1+x_0/2+x_1/4+x_2}) ]$
Let $f:~ [0,1]^3 \rightarrow \mathbb{R}$ be
$$
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
0
answers
96
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
1
answer
137
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
3
answers
204
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
0
answers
68
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
1
answer
155
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
1
answer
124
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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
1
answer
198
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
0
answers
567
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
1
answer
231
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
2
answers
353
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
1
answer
429
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 ...
2
votes
0
answers
58
views
Convex solutions of linear hyperbolic PDEs in a planar domain
Consider a linear homogeneous 2nd-order PDE in a convex planar domain $\Omega$ :
$$a(x,y)\frac{\partial^2u}{\partial x^2}+2b(x,y)\frac{\partial^2u}{\partial x\partial y}+c(x,y)\frac{\partial^2u}{\...
1
vote
0
answers
59
views
Condition on $f := (I-\delta P)^{-1}g$ to ensure convexity of $f$?
Consider the following extension of the notion of convexity of continuous functions to functions defined over $\mathcal{I} = \{0,\ldots,n \}$ — that is, to vectors: $f$ is convex if for any $\alpha \...
11
votes
2
answers
520
views
Convex hull of the Stiefel manifold with non-negativity constraints
Consider the Stiefel manifold
$$\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( ...
1
vote
0
answers
613
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
0
answers
62
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
0
answers
150
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,...