Questions tagged [convexity]
For questions involving the concept of convexity
636 questions
-1
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0
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24
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A question on Ibragimov's theorem on strong unimodality
I am not a mathematics student and unfortunately have some confusion about a (well-known) theorem about strong unimodality of distributions. First of all let me clarify some terminologies and then ask ...
- 49
0
votes
0
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21
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Largest inscribed parallelepiped of the convex set defined by partial sum of Fourier series
Let $\mathcal{X}$ be the set consisting of all $(2n+1)$-dimensional real vectors $\mathbf{x}=\left( a_0,a_1,\ldots,a_n,b_1,\ldots,b_n\right)^{\intercal}$ satisfying
$$
\left| f_{\mathbf{x}}(t) \right|...
- 550
11
votes
2
answers
433
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On the convex cone of convex functions
$\newcommand\R{\Bbb R}$Let $F$ be the set of all functions of the form $\max(a,b,c)$, where $a,b,c$ are affine functions from $\R^2$ to $\R$ and the maximum is taken pointwise. Let $G$ be the set of ...
- 128k
2
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0
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95
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Why cannot we adapt Barvinok type counting techniques to general convex integer programs?
Decision problems in Integer Linear Programming have Lenstra type algorithms (https://www.math.leidenuniv.nl/~lenstrahw/PUBLICATIONS/1983i/art.pdf) have been generalized to convex integer program ...
- 13.9k
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52
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What are the injective embeddings of R^d into the cone of (semi-) positive definite matrices of dimension d?
How can we characterize the set of all injective functions from $\mathbb{R}^d$ to the set of all symmetric positive definite matrices of dimension d?
- 109
2
votes
0
answers
58
views
An s-convex function lying between two convex functions
Let $f: \mathbb R_{+} \to \mathbb R_{+}$ be an $s$-function in the second sense, i.e.,
$$ f(\lambda x +(1-\lambda)y) \leq \lambda^s f(x) +(1-\lambda)^s f(y)$$ for every $\lambda \in (0,1)$. Assume ...
- 55
1
vote
0
answers
30
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Generalization of subadditivity analogous to quasiconvexity, and variants
I am curious if there are natural generalizations of subadditivity which have been studied in the past or have been stated in the literature? I (and people that I have talked to) have not had much ...
- 21
4
votes
1
answer
417
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Maximum and concavity of function
Let
\begin{align}
G(x_1,x_2,x_3)=\int_0^{2\pi}\int_0^{2\pi}\int_0^{2\pi} \frac{d\theta_1 d\theta_2 d\theta_3}{(2\pi)^3} p(\vec{x},\vec{\theta}) \log\left( p(\vec{x}, \vec{\theta})\right)
\end{...
- 231
2
votes
0
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108
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Reference request: books on convex analysis / geometry
I am interested in convex geometry and analysis, especially in its connections with high dimensional probability theory.
I was reading the book by Pisier, The volume of convex bodies and Banach space ...
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0
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0
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17
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Convexity property of a divergence measure with constant factor
I have a divergence measure (D) that satisfies the following property for any functions $f_1$, $f_2$, and $g$:
$$D(\lambda f_1 + (1 - \lambda) f_2 \| g) \leq \frac{c_2}{c_1} \left( \lambda D(f_1 \| g) ...
- 101
3
votes
0
answers
77
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Is the norm of first or second level of of signature a convex function?
I understand this is not a research level question but I really want to know, would anyone please help.
This question is related to the signatures that arises in rough path theory. https://en....
- 495
4
votes
1
answer
101
views
Limits along lines for the gradient of a convex function
It is easy to see that if a function $f: \mathbb{R} \to \mathbb{R}$ is strictly convex, $C^1$ and $f'$ has bounded image, then as $t\to \infty$ the limit
$$
\lim_{t\to\infty} f'(t) = \lim_{t\to\infty} ...
- 119
9
votes
2
answers
424
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Is there a path-connected, "anti-convex" subset of $\mathbb R^2$ containing $(\mathbb R\smallsetminus \mathbb Q)^2$?
This question was firstly asked in mathematics stack exchange. Getting no answer, I copied it to here.
For a vector space $V$ over $\mathbb R$, I say a subset $S$ of $V$ is "anti-convex" if $...
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6
votes
0
answers
48
views
Strengthening the Kovner-Besicovich theorem: Does every unit-area convex set in the plane contain a centrally symmetric hexagon of area $2/3$?
The Kovner-Besicovich theorem states that every convex set $S$ in the plane contains a centrally symmetric subset $C$ of at least $2/3$ the area of $S$, and that this bound is sharp for triangular $S$....
- 3,068
0
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1
answer
59
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Do separable cubic constraint and separable quartic constraint SOCP presentable?
I am an engineer who is doing some network modeling and optimization. During my work, I was running into a case that is quite strange. The problem that I am trying to solve seems to be convex and it ...
7
votes
2
answers
345
views
Integral means vs infinite convex combinations
Let $(X,\mathcal A, \mu)$ be a probability space, $\mathbb E$ a Banach space, and $f:X\to\mathbb E$ a Bochner integrable function.
Does there exist a sequence $(x_k)_{k\ge 1} $ in $X$, and a ...
- 60.6k
0
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0
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40
views
Construct compact submanifold containing non-compact Nash embedded submanifold
$$
\newcommand{\R}{\mathbb{R}}
\newcommand{\geu}{g_{\text{Eu}}}
\newcommand{\X}{\mathcal{X}}
\newcommand{\iX}{\mathring{\X}}$$
Let $\X$ be a closed bounded convex set in some Euclidean space. Its ...
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2
votes
1
answer
143
views
Proving convexity of the expected logarithm of binomial distribution
I would like to prove that the following function, for an arbitrary integer $n$:
\begin{equation}
\begin{split}
f(x) & =x\cdot E \ \log(1+\text{Binomial(n,x)}) \\
& = x \cdot \sum_{k=0}^{n} \...
- 23
1
vote
0
answers
30
views
Finite right-triple convex sets in planes
Let $\mathcal{S}$ be a set of points in $\mathbb{R}^2$. We say that $\mathcal{S}$ is right-angle convex, if for any two distinct points $P,Q\in \mathcal{S}$ there always exists another point $R\in \...
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2
votes
1
answer
172
views
Does there exists an example of a Banach space that is compactly LUR; but not LUR
We know that a Banach space $X$ is locally uniformly rotund(LUR) if for $x, x_n \in S_X$ with $\Vert x+x_n \Vert \to 2$, we have $x_n \to x$. In the same case, if $(x_n)$ has a convergent subsequence, ...
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2
votes
1
answer
183
views
Convexity of a function
Let: $F_{j+1,y}(s)$ be the cumulative distribution function of a binomial distribution with mean $y$, $j+1$ independent trials considered for $s$ successes. Is it possible to show in any way that:
$\...
4
votes
1
answer
139
views
Characterization of convexity by connectedness of hyperplane sections
Let $S$ be a smooth closed connected embedded hypersurface in $\mathbb R^n$.
Is it true that $S$ is convex, i.e. is a boundary of a convex set, if and only if any section of $S$ by a hyperplane is ...
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3
votes
0
answers
208
views
Reference request: Carathéodory-type theorem for convex hulls of closed sets
I'm looking for a reference for the following theorem.
Theorem Let $X$ be a closed subset of $\mathbb{R}^N$, and let $a$ be a point of its convex hull $\operatorname{conv}(X)$. Then there exist ...
- 27.7k
3
votes
1
answer
296
views
Does this condition characterise intervals, among subsets of the real line?
For a real number, $c\in \left]0,1\right[$, consider the following property $\mathbf(\mathbf P_c\mathbf)$ of subsets $A$ of $\mathbb R$:
$\mathbf(\mathbf P_c\mathbf)$ For every bounded set $B\subset \...
- 60.6k
0
votes
0
answers
49
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Conditions on symmetric $3 \times 3$ matrices to satisfy the convex equality for cofactor and determinant
Given any $3\times 3$ finite set of symmetric matrices $A_i$ and positive real $a_i$ such that $\sum_ia_i=1.$ Is there any equivalent condition to the existence of skew symmetric matrices $X_i$ such ...
1
vote
1
answer
98
views
If $|P|<\infty$ and $C=P\cap\partial(\textrm{Conv}(P))$, then $P\subset\textrm{Conv}(C)$?
That is, if $P$ is a finite set, and $C$ is the set of points in $P$ which lie on the boundary of the convex hull of $P$, then is $P$ contained in the convex hull of $C$?
It seems true intuitively. In ...
19
votes
0
answers
553
views
Talagrand's "Creating convexity" conjecture
We say a subset $A$ of $\mathbb{R}^N$ is balanced if
\begin{equation}
x \in A, \lambda \in [-1,1] \implies \lambda x \in A.
\end{equation}
Given a subset $A$ of $\mathbb{R}^N$, we write
\begin{...
- 484
4
votes
1
answer
321
views
Sub-Gaussian random variables and convex ordering
Suppose that $X$ is a $1$-sub-Gaussian real-valued random variable, i.e. for all $t \in \mathbf{R}$, it holds that $\log \mathbf{E} \exp \left( t X \right) \leqslant \frac{1}{2} t^2 $.
Does there ...
- 801
1
vote
1
answer
142
views
Foliation of spaces
It turns out that a very important idea to derive properties for a bigger space is to try to foliate the space, derive the same property for each leaf and patch everything up to get the desired ...
- 203
1
vote
0
answers
95
views
Detecting points inside the convex hull with inner products
Given a finite set $P$ of $n\gt d+1$ points in $d$-dimensional euclidean space.
Under the assumption that the points of $P$ are in general position in the sense that $\lbrace p_{i_1},\dots,p_{i_d}\...
- 13.2k
3
votes
2
answers
293
views
On convergence of convex-concave functions
Let $(f_n)$ be a sequence of twice differentiable functions on $\mathbb R$ such that for each $n$ there exists some $x_n\in\mathbb{R}$ such that:
$f_n$ is strictly convex on $(-\infty,x_n)$,
$f_n$ is ...
- 128k
1
vote
1
answer
300
views
Convergence of concave/convex function
Let assume that you have a sequence of twice differentiable functions $(f_n)_{n\in\mathbb{N}}\in\mathscr{C}^2(\mathbb{R})^{\mathbb{N}}$. Let suppose that for each $f_n$, it exists a $x_n\in\mathbb{R}$ ...
- 393
5
votes
1
answer
455
views
Is there a Borel measurable $f:\mathbb{R}^d \to \mathbb{R}^d$ such that $f(x) \in \partial \varphi (x)$ for all $x$?
Let $\varphi: \mathbb{R}^d \to \mathbb{R}$ be a convex function. The subdifferential of $f$ at $x$ is defined as
$$
\partial \varphi (x) := \{z \in \mathbb{R}^d : \varphi(y) \geq \varphi(x) + \langle ...
- 825
1
vote
1
answer
144
views
Is the space of bounded $\psi_\infty$ Orlicz norm random variables equal to $L^\infty$?
Let $\psi_\alpha(x) = \exp(x^\alpha)-1$ for $\alpha\geq 1$.
Define
$$
\psi_\infty(x) = \begin{cases}\infty & x>1\\1& x = 1\\ 0 & x <1
\end{cases}
$$
to be such that for any $x>0$ $...
- 2,183
2
votes
1
answer
170
views
Log-concavity of the difference of the second anti-derivative of Gaussians
I would like to prove the following but I couldn't manage to do it. Let $a>b>0$ be two real numbers. Let $f$ be the function defined as:
$$\forall \sigma>0, ~\forall x\in\mathbb{R},~f_\sigma(...
- 393
3
votes
1
answer
120
views
How to establish regions of convexity/concavity of a ratio of exponential polynomials?
Problem:
Let $f\colon \mathopen[0,1\mathclose] \to \mathbb{R}$ be defined as
$$
f(x) = \frac{e^{\rho x}-1}{e^{\rho x}-1+e^{\rho (1-\gamma x)}-e^{\rho (1-\gamma) x}}
$$
where $\rho$ and $\gamma$ are ...
- 33
2
votes
1
answer
183
views
Exponential optimization problem
\begin{eqnarray}
\arg\max_{k}\sum_{i=1}^{p}\sum_{j=1}^{p}\exp\left(-{\frac{\left(X(i,j)-{U_k}(i,j)\right)^2}{2}}\right),\:\: k=0,\dots,p
\end{eqnarray}
where $X$ and $U_k$ are the $p\times p$ matrices,...
- 21
0
votes
1
answer
113
views
On the extreme points of two convex sets
Let $A$ and $B$ be two compact convex sets (which may be assumed to be polytopes) in $\mathbb R^n$ such that $A\cap B\ne\emptyset$. Is it then always true that either $A\cap\text{ext}B\ne\emptyset$ ...
- 128k
2
votes
0
answers
164
views
Log Sobolev inequality for log concave perturbations of uniform measure
Suppose $\Omega$ is a convex bounded open set of $\mathbb{R}^n$ (I would be happy with just $\Omega$ as the $n$-dimensional cube). Let $\mu$ be the uniform measure on $\Omega$ and consider the ...
- 873
5
votes
1
answer
179
views
On the property P in the Whitney extension theorem
Let $D$ be a possibly unbounded domain in $\mathbb{R}^d$, $d \ge 2.$
We say that $D$ has the property P if there exists $C>0$ such that such that any pair of points $x,y \in D$ can be joined by a ...
- 721
1
vote
1
answer
147
views
Link between asymptotic cone and the boundary of a convex set
For $n\geq 1$, let $f\in\mathcal{C}^2(\mathbb{R}^n ; \mathbb{R})$ such that $f^{-1}(\mathbb{R}_-^*)\neq\varnothing$ and $\forall x\in\mathbb{R}^n$: $\nabla f(x)\neq 0$ and $\mathcal{Hess}f(x)$ is ...
- 449
18
votes
3
answers
998
views
Convex functions in convex sets
Suppose $\Omega \subset \mathbb{R}^n$ is some bounded, convex set. For which domains $\Omega$ is it true that for every convex function $f:\Omega \rightarrow \mathbb{R}$ the average of the function in ...
3
votes
2
answers
350
views
Subdifferential of a convex function admits a continuous selection
Let $F$ be a continuous convex function on $\mathbb{R}^n$.
If the subdifferential $\partial F(x)$ of $F(x)$ admits a continuous selection, for every $x \in \mathbb{R}^n$, does it mean that $F$ is ...
- 33
2
votes
1
answer
106
views
Submodularity of a particular function derived from a convex function?
Consider a convex function $f : \mathbb{R}^d \to \mathbb{R}$. Define now the set-input function $g_f : 2^{[d]} \to \mathbb{R}$ as follows,
\begin{align}
g_f(S) = \min \left\{ f(x) : x \in \mathbb{R}^d ...
- 297
4
votes
1
answer
425
views
An exercise on log-concave random variable on the real line
Let $X$ be a real random variable with log-concave density $f$. Assume that $E(X) =0$ and $E(X^2)=1$.
Show that there is a universal (independent of $X$) constant $c>0$ such that:
$$P(X\in[-1/2;0])\...
- 245
2
votes
1
answer
599
views
Tangent cone of a closed convex cone
Let $K \subset \mathbb{R}^n$ be a closed convex set. Given a point $u \in K$, the tangent cone of $K$ at $u$ is defined as (or characterized by)
$$
T_K(u) := \mathrm{cl}(\left\{ t (v - u) \mid v \in K,...
- 163
0
votes
0
answers
141
views
Does there exists a smooth reflexive infinite dimensional Banach space that is not strictly convex
It is known that in a reflexive Banach space, if the norm is strictly convex, then its dual will be smooth Banach space, and if the norm is smooth, then the dual norm is strictly convex.
We can find ...
- 85
0
votes
1
answer
138
views
Smoothness of a Hilbert space under an equivalent norm
Let us take the Hilbert space $l_2$ with an equivalent norm
$\Vert x \Vert = \max \{2 \Vert x \Vert_1, \Vert x \Vert_2 \}$, where $\Vert x \Vert_1 =( \sum_{n=2}^\infty x_n^2 )^{\frac{1}{2}}$ and $\...
- 85
5
votes
2
answers
204
views
Convex hull of bivariate normal random points
Let $n > 1$, and $p_1, \ldots, p_n \in \mathbb{R}^2$ be iid random points following a bivariate normal distribution (doesn't matter which one), and let $X_n$ be the number of vertices of convex ...
- 5,590
1
vote
2
answers
126
views
Establishing quasiconcavity
Let $f:\mathbb{R}_+\rightarrow\mathbb{R}_+$ be twice differentiable quasi-concave function satisfying $f(x)>0,\forall x \in \mathbb{R}_+$. Let $g:\mathbb{R}_+\rightarrow\mathbb{R}_+$ be a positive, ...
- 67