Questions tagged [convexity]
For questions involving the concept of convexity
625
questions
4
votes
1
answer
278
views
Intrinsic definition of a cone in a normal fan
Let $P\subseteq \mathbb{R}^n$ be a full dimensional polytope. Let us assume that $P$ has a facet description with the following inequalities:
$$ \left<x,u_F\right> \geq -a_F$$
where $u_F\in \...
0
votes
0
answers
34
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 ...
2
votes
1
answer
97
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} \...
1
vote
0
answers
26
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 \...
67
votes
3
answers
11k
views
Nonconvexity and discretization
Edit: Here's a more down-to-earth, and somewhat weakened, but I believe still nontrivial, version of the main theorem.
Prototypical nonconvex spaces are $\ell^p$-spaces for $0<p<1$, say $\ell^p(\...
2
votes
1
answer
156
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, ...
2
votes
1
answer
105
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
128
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 ...
0
votes
2
answers
218
views
Decreasing magnitude of spherical centroid
Let $\sigma$ be the uniform measure on $\mathbb{S}^{d-1}\subset \mathbb{R}^d$. For any region $R\subset \mathbb{S}^{d-1}$, let $X_R$ be a random variable which is uniformly distributed across $R$. We ...
3
votes
0
answers
110
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 ...
3
votes
1
answer
432
views
Generating uniquely $k$-optimal point sets
This question is motivated by the observation that finding an optimal tour through a set of points in the Euclidean plane is especially simple, if the points are in convex configuration and, that the ...
3
votes
1
answer
285
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 \...
0
votes
0
answers
46
views
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
118
views
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 $i=1,2$. ...
5
votes
2
answers
168
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 ...
1
vote
1
answer
109
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,...
5
votes
1
answer
366
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
$$
\partial \varphi (x) := \{z \in \mathbb{R}^d : \varphi(y) \geq \varphi(x) + \langle ...
1
vote
1
answer
289
views
How to prove the convexity of a simple function involving a ratio of two polygamma functions?
Let
\begin{equation*}
\Gamma(z)=\int_0^{\infty}t^{z-1}\textrm{e}^{-t}\textrm{d}t, \quad \Re(z)>0
\end{equation*}
and
$$
\psi(z)=[\ln\Gamma(z)]'=\frac{\Gamma'(z)}{\Gamma(z)}.
$$
In the literature, ...
1
vote
1
answer
90
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 ...
13
votes
0
answers
301
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{...
3
votes
1
answer
226
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 ...
1
vote
1
answer
112
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 ...
1
vote
0
answers
62
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}\...
7
votes
1
answer
1k
views
Why are all these families of polynomials finally log-concave?
This started when I was examining certain families of unimodal polynomials, i.e. $\sum_{k=0}^n a_kx^k$ where $a_0\le a_1\le\cdots \le a_k\ge\cdots \ge a_n$.
(Notation: in the following, the $a_k$ ...
3
votes
0
answers
491
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 ...
3
votes
2
answers
252
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 ...
1
vote
1
answer
220
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}$ ...
37
votes
0
answers
1k
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 ...
0
votes
0
answers
38
views
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^{...
6
votes
2
answers
300
views
For most directions does the supporting hyperplane meeting a bounded convex set meet it in one point?
Let $C\subseteq \mathbb R^n$ be non-empty, convex and compact. For $v\in S^{n-1}$, let $H_v$ be the supporting hyperplane in the direction of $v$ (i.e., $H_v$ is the boundary of the smallest closed ...
0
votes
0
answers
62
views
Looking for a homogeneous function with some properties
I'm looking for a 1-homogeneous function $\pi \colon \mathbb{R}^n_{\geq 0} \to \mathbb{R}$ satisfying the following properties:
1) $\pi$ is not concave. This is equivalent to the fact that there ...
1
vote
1
answer
113
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}
$$
to be such that for any $x>0$ $...
3
votes
1
answer
106
views
How to establish regions of convexity/concavity of a ratio of exponential polynomials?
Problem:
Let $f\colon \mathopen[0,1\mathclose] \to \mathbb{R}$ be defined as
$$
f(x) = \frac{e^{\rho x}-1}{e^{\rho x}-1+e^{\rho (1-\gamma x)}-e^{\rho (1-\gamma) x}}
$$
where $\rho$ and $\gamma$ are ...
2
votes
1
answer
140
views
Log-concavity of the difference of the second anti-derivative of Gaussians
I would like to prove the following but I couldn't manage to do it. Let $a>b>0$ be two real numbers. Let $f$ be the function defined as:
$$\forall \sigma>0, ~\forall x\in\mathbb{R},~f_\sigma(...
18
votes
3
answers
915
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 ...
7
votes
1
answer
1k
views
Convex conjugate of relative entropy
The convex conjugate of a function, say, $f:X\to \mathbb{R}$ is a function $f^*:X^*\to \mathbb{R}$ defined as
$$f^*(x^*):=\sup_{x\in X} ~\langle x, x^*\rangle-f(x),$$ where $X^*$ is the topological ...
7
votes
2
answers
438
views
Proving the set $\left\lbrace \frac{(x + y)^2}{\sqrt{y}} \leq x - y + 5, y > 0 \right\rbrace$ is convex
I have recently picked up a course on Convex Analysis in my spare time, but feel I'm not quite up to speed with the 'tricks' for proving a set is convex.
I have managed to prove this by moving all ...
34
votes
16
answers
7k
views
Generalizations of the Birkhoff-von Neumann Theorem
The famous Birkhoff-von Neumann theorem asserts that every doubly stochastic matrix can be written as a convex combination of permutation matrices.
The question is to point out different ...
0
votes
1
answer
89
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
103
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 ...
1
vote
1
answer
129
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 ...
5
votes
1
answer
170
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 ...
4
votes
2
answers
11k
views
Convexity of a minimum function
I was reading a proof of $9g-9$ theorem which states that $9g-9$ length parameters are sufficient the parametrize the Teichmuller space of a closed surface of genus $g$. The proof uses the following ...
0
votes
0
answers
80
views
Generalization of Kakutani-Ky Fan Theorem without convexity assumptions
Crossposted at Mathematics SE
I'm wondering if there exists some extension of the Kakutani-Ky Fan Theorem
Theorem. Let $K$ be a nonempty, compact and convex subset of a locally convex space $X$. ...
3
votes
2
answers
178
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 ...
2
votes
1
answer
99
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 ...
4
votes
1
answer
416
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:
$$P(X\in[-1/2;0])\...
0
votes
0
answers
62
views
Representation of concave point-to-set maps
Given a point-to-set map $C: X \rightrightarrows Y$ defined by some vector valued-function $\mathbf{g}: X \times Y \to \mathbb{R}^n$ such that $C(x) \doteq \{y \in Y | g_1(x,y), …, g_n(x,y) \geq 0 \}$,...
2
votes
1
answer
293
views
Tangent cone of a closed convex cone
Let $K \subset \mathbb{R}^n$ be a closed convex set. Given a point $u \in K$, the tangent cone of $K$ at $u$ is defined as (or characterized by)
$$
T_K(u) := \mathrm{cl}(\left\{ t (v - u) \mid v \in K,...
0
votes
0
answers
112
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 ...