Questions tagged [convexity]

For questions involving the concept of convexity

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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 ...

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