# Questions tagged [convexity]

For questions involving the concept of convexity

628
questions

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### Convex covering of topological space

Let $V$ be the set of all Riemann integrable functions. $V$ forms a vector space with respect to general function addition and scalar multiplication. Let $X \subset V$ is the set of positive and ...

4
votes

1
answer

91
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### 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} ...

9
votes

2
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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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### 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 $S$....

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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
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2
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323
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### 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 ...

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39
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### 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 ...

2
votes

1
answer

118
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### 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} \...

1
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0
answers

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### 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 \...

2
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### 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, ...

2
votes

1
answer

135
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### 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

134
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### 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 ...

3
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### 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 ...

3
votes

1
answer

287
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### 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 \...

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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

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### 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 ...

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### 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{...

3
votes

1
answer

257
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### 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 ...

1
vote

1
answer

122
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### 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 ...

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### 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}\...

3
votes

2
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272
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### 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 ...

1
vote

1
answer

257
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### 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 $x_n\in\mathbb{R}$ ...

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### Is there any generalization of the convexity of $t^{-p}$ for $p > 0$ for real symmetric positive definite matrices?

Let $p > 0$. On the positive reals, $t \mapsto t^{-p}$, is a convex function, as can be seen easily by a plot or differentiation.
However, unfortunately, unless $p \in (0, 1]$, the map $f_p(X) = X^{...

3
votes

0
answers

495
views

### Prove concavity of real valued function on the non-negative real axis

Fix $\alpha >0$ and define $f_{\alpha}(x) := \ln(\Phi(\alpha-x)-\Phi(-\alpha-x))$, where $\Phi(x)$ is the normal cumulative density function. For some research, I am trying to verify that the ...

5
votes

1
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407
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### 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 ...

1
vote

1
answer

136
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### 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
votes

1
answer

157
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### 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(...

3
votes

1
answer

112
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### 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 ...

1
vote

1
answer

147
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### 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,...

0
votes

1
answer

98
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 $A\cap\text{ext}B\ne\emptyset$ ...

2
votes

0
answers

124
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 ...

5
votes

1
answer

172
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 ...

1
vote

1
answer

137
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### 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 ...

18
votes

3
answers

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### 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
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213
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### 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 ...

2
votes

1
answer

101
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### 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 ...

4
votes

1
answer

419
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])\...

2
votes

1
answer

410
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### 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,...

0
votes

0
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### 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 ...

0
votes

1
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137
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### 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 $\...

5
votes

2
answers

179
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### 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 ...

1
vote

2
answers

108
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### 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, ...

4
votes

1
answer

143
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### Approximation of convex bodies by polytopes corresponding to smooth toric varieties

Let $P\subset \mathbb{R}^n$ be an $n$-dimensional polytope with rational vertices. There is a well known construction which produces an $n$-dimensional algebraic variety $X_P$ called toric variety. In ...

2
votes

1
answer

160
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### Higher-order convexity

Let $f \in C^\infty(\mathbb R)$.
$f^{(1)}=f'\geq 0$ iff $\forall (a,b) \in\mathbb R^2,a\leq b$ then $f(a) \leq f(b)$
$f^{(2)}=f''\geq 0$ iff $\forall (a,b)\in\mathbb R^2,\forall t\in [0,1], f(ta+(1-t)...

1
vote

1
answer

108
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### Any example of a multi-valued monotone maximal operator without subdifferential?

Is there an example of a multivalued maximal monotone operator that is not the convex subdifferential of a proper convex lower semicontinuous? Besides, among these type of operators, are there any ...

11
votes

2
answers

656
views

### Existence of an open convex set

Let $T$ be a normed vector space, $K\subseteq T$ compact and convex and $O\subseteq K$ convex and open in $K$ (i.e. open w.r.t. the subspace topology of $K$ inherited by $T$).
Can we find an open set $...

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vote

1
answer

891
views

### Is a Lipschitz continuous gradient equivalent to this condition?

I know if a function $f: \mathbb{R}^n \to \mathbb{R}$ is $L$-smooth, i.e. its gradient $\nabla f$ is $L$-Lipschitz continuous, then it satisfies the following inequality for any $x, x_0 \in \mathbb{R}^...

4
votes

0
answers

132
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### A Lipschitzian's condition for the measure of nonconvexity

I'm actually working on the measure of nonconvexity and its application. Especially, the Eisenfeld–Lakshmikantham MNC defined - in a Banach space - by:
$$\alpha(A)=\sup_{b\in\overline{\operatorname{...

5
votes

1
answer

148
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### On existence of a concave function

Let $a$ be a strictly positive $C^\infty$ smooth function on the unit interval. Does there exist a strictly positive $C^\infty$ smooth function $f$ on $I$ such that
$$ f’’(x) \leq 0\quad \text{and} \...

1
vote

1
answer

147
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### Comparison of solutions of Hamilton-Jacobi equations with different initial conditions

Consider a Hamilton-Jacobi equation:
$$u_{t} + f(u_{x}) = 0 \quad (x,t) \in \mathbb{R}\times [0,+\infty)$$
with two possible initial conditions $u(x,0) = g_{i}(x)$ for $x \in \mathbb{R}$ and $i=1,2$. ...