Questions tagged [convexity]

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$ h $ is an affine function?

Let $X$ be a separable Banach space, the associated dual space is denoted by $X^*$ and the usual duality between $X$ and $X^*$ by $\langle , \rangle$. For $C$ nonempty weakly compact convex subsets of ...
2
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1answer
55 views

Naming convention: looking for better terminology for “centrally symmetric smooth strictly convex bodies”

I have recently found myself researching a certain type of convex body in $\mathbb{R}^2$, namely centrally symmetric smooth strictly convex bodies. Instead of repeating such a sentence repetitively I ...
5
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1answer
172 views

a square root inequality for symmetric matrices?

In this post all my matrices will be $\mathbb R^{N\times N}$ symmetric positive semi-definite (psd), but I am also interested in the Hermitian case. In particular the square root $A^{\frac 12}$ of a ...
4
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1answer
114 views

Is there a non-convex function with non-decreasing average rate of change?

$\newcommand{\R}{\mathbb R}$ Let $f$ be a function from $\R$ to $\R$. It is said that $f$ is midpoint-convex if for any real $x$ and $y$ we have $f(x-y)+f(x+y)\ge2f(x)$ or, equivalently, $f(x+y)-f(x)\...
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16 views

Order-relational conditions for 4 points being in convex configuration

In the euclidean plane a simple sufficient condition for 4 points being in convex configuration is as follows: if the points are $\lbrace A,\,B,\,C,\,D\rbrace$ of which $\lbrace A,\,B,\,C\rbrace$ are ...
2
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0answers
36 views

A complete metric space with some convex-type property

Let $(X,d)$ be a complete metric space with this property: for each $x∈X$, $r>0$ and $y∈X$ with $d(x,y)<r$, there exists $z∈X$ such that $d(x,y)+d(y,z)=d(x,z)=r$. I want to know if the family ...
1
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1answer
75 views

Gradient flows: convex potential vs. contractive flow?

Take a $\mathcal C^2$ potential $V:\mathbb R^d\to \mathbb R$, and assume that it is bounded from below (say $\min V=0$ for simplicity, so that $V\geq 0$). Consider the autonomous gradient-flow $$ \dot ...
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57 views

Effective radius of section of a convex set compared to that of the convex itself

The effective radius $er(A)$ of a $n$-solid $A$, is defined by Schramm (see the question by Gil Kalai Volumes of Sets of Constant Width in High Dimensions) to be the radius of the $n$-ball that has ...
2
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1answer
276 views

real analytic function with given shape

I am looking for a 5 parameter family of analytic functions $f:[0,1]\to R$ such that (0) $f$ has zeros at $0,p,1$. (1) $f$ is convex in $[0,p]$ and concave in $[p,1]$. (2) The five parameters, $p$ ...
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1answer
70 views

Superdifferentiable and subdifferentiable at $x$ implies differentiable at $x$

Let $(M,g)$ be a compact connected smooth Riemannian manifold without boundary and let $\phi: M \rightarrow \mathbb{R}$ be a function on $M$. We say that $\phi$ is superdifferiantiable at $x$ with ...
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1answer
163 views

Conditions such that norm of matrix vector can be written as the derivative of the norm of the vector for some convex fonction

Problem statement: Let $A$ be a matrix $\mathbb{R}^{d \times d}$, I want to find some conditions on $A$ such that there exists a differentiable convex function $f: \mathbb{R_{+}} \rightarrow \mathbb{...
2
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1answer
119 views

Is the optimum of this problem convex in the constraint parameter?

Let $f:\mathbb R^+ \to \mathbb R$ be a smooth function, satisfying $f(1)=0$, and suppose that $|f|$ grows with the distance from $1$: $|f(x)|$ is strictly increasing when $x \ge 1$, and strictly ...
4
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4answers
222 views

algorithm for convex $C^2$ interpolation

Let $x_0<x_1<\ldots<x_n$ and $f_0,f_1,\ldots,f_n$ be real numbers and $$s_i=(f_i-f_{i-1})/(x_i-x_{i-1}),~~~c_i=(s_{i+1}-s_i)/(x_{i+1}-x_{i-1}).$$ If $f$ is a convex function defined on $[x_0,...
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0answers
27 views

Extended-value subgradients

(I am not an expert in convex analysis, as may become clear.) Let $R$ be the extended reals, $R = \mathbb{R} \cup \{\pm \infty\}$. Standard texts define the subgradient of a convex function $f: \...
3
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1answer
132 views

Geodesic convexity and the Geometric Hessian

This is an elementary question in differential geometry. We know that for a smooth real-valued function $f$ defined on an open geodesically convex set of a Riemannian manifold $ \mathcal{X} \subset \...
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77 views

Finding a specific solution to $X^T\Sigma X = D$

I'm looking to solve for a specific $X$ in the following equation: $$X^T\Sigma X = D,$$ where $\Sigma \succ 0$, $D$ is a diagonal matrix with strictly positive entries, and all matrices are square. It ...
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1answer
331 views

Aleksandrov's proof of the second order differentiability of convex functions

Aleksandrov [A], proved a remarkable property of convex functions. Theorem. If $f:\mathbb{R}^n\to\mathbb{R}$ is convex, then for almost every $x\in\mathbb{R}^n$ there is $Df(x)\in\mathbb{R}^n$ and ...
6
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1answer
249 views

A conjecture (or theorem?) on unit vectors in a Euclidean space

I have heard (if I am not mistaken) that there exists the following conjecture (or theorem?). Let $u_1,\dots,u_n$ be unit vectors in an $n$-dimensional Euclidean vector space. Then there exists ...
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2answers
700 views

Reference to a conjecture on unit vectors in Euclidean space

I have heard that there exists the following conjecture (if I am not mistaken). Let $u_1,\dots,u_n$ be unit vectors in an $n$-dimensional Euclidean vector space. Then there exists another unit vector ...
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0answers
105 views

A property of $f(x)=x^{4(1-x)^2}+(1-x)^{4x^2}$

Let $0<x<0.5$ or $0.5<x<1$ defines the following function : $$f(x)=x^{4(1-x)^2}+(1-x)^{4x^2}$$ Claim : The function $g(x)=\exp(|1-f(x)|)-1$ is a logarithmically concave function Or :...
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0answers
77 views

Isometries between two convex bodies [closed]

Let $A,B$ be two convex compact subsets with non-empty interior in a Euclidean space $\mathbb{R}^n$. Let $f\colon A\to B$ be a bijective isometry between them. Does there exist an isometry $F\colon ...
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58 views

Linearly independent support vectors of a convex set

Let $\Omega\subset\mathbb{R}^n$ a compact strictly convex set containing $0$ in its interior and let $k\leq n$. Given a vector $x\neq 0$ in $\mathbb{R}^n$ a supporting vector $\xi_x$ in the ...
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0answers
105 views

On intrinsic volumes

Let $\Gamma$ be a convex polytope in $\mathbb R^n$. The $k$-th intrinsic volume of $\Gamma$ is the number $$ \text{v}_k(\Gamma)=\sum_{\Delta\in{\mathcal B}(\Gamma,k)}\text{vol}_k(\Delta)\psi_\Gamma(\...
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0answers
220 views

Open convex hull of a closed set

Let $X$ be a closed set in a Euclidean space of finite dimension and suppose that its convex hull $H$ is open. I can prove that, in this case, $H$ is a Cartesian product of a line with an open convex ...
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1answer
282 views

Does midpoint-convex imply rationally convex?

Say that a function $f$ defined on $\mathbb{Q}^n$ is midpoint convex if $f((x+y)/2) \le (f(x) + f(y))/2$. Say that it is rationally convex if, for $\lambda \in \mathbb{Q} \cap [0,1]$ and $\bar \lambda ...
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0answers
53 views

Proximity operator composition

Suppose that $f,g$ are convex and lower-semi continuous functions from $\mathbb{R}^d$ to $\mathbb{R}$ such that $f\circ g$ is lsc and convex. I know that the proximity operators $\operatorname{Prox}...
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1answer
137 views

Are closed convex subsets of a Banach space weakly closed without the axiom of choice?

It is a well-known fact that closed convex sets in Banach spaces are weakly closed. The common proof is based on the Hahn-Banach theorem that uses the axiom of choice. Is there any proof of this fact ...
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1answer
218 views

Expectations, double integrals and Jensen's inequality

$\def\anonfunc#1{#1(\cdot)}$Consider two random variables distributed $v\sim \anonfunc G$ and $c \sim \anonfunc F$ with pdfs $\anonfunc g$ and $\anonfunc f$. Let the supports of $c$ and $v$ be $[x,y]$....
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0answers
66 views

Sufficient condition for convex conjugate to be second-order differentiable

Let $f:\mathbb{R}^n\to (-\infty,\infty]$ be a convex lower-semicontinuous function, we then define its conjugate by $$ f^*(y)=\sup_{x\in \mathbb{R}^n}\{x^Ty-f(x)\}. $$ Then there exist well-known ...
3
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1answer
153 views

Quotient space of a locally uniformly rotund space

If $X$ is a uniformly rotund space , then for any closed subspace $M$ of $X$, $X/M$ is uniformly rotund. Does this hold for a locally uniformly rotund space? That is if $X$ is locally uniformly ...
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2answers
193 views

Goldowsky-Tonelli theorem for upper semi continuous function

Let $f:(0, \infty) \rightarrow \mathbb{R}$ be a convex and continuous function. We know that $\partial^{e} f(. )$(either right or left derivative) is non-decreasing and upper semi-continuous function. ...
3
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1answer
137 views

Is there a definition for “convexity” of spatial (non-planar) polygons? [closed]

I was thinking that there should exist a definition for "convexity" of spatial polygons. A planar convex quadrilateral that has one vertex moved (perpendicularly) out of the plane should continue to ...
4
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1answer
92 views

What is the probability of an empty convex $k$-gon among many given points?

Given a finite number of points in the plane in general position, call a convex subset empty if its hull doesn't contain any other of the points. For a big number $n$ of randomly distributed ...
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0answers
60 views

Convexity of conditional relative entropy for Markov distributions

Consider two Markov processes $p$ and $q$. The conditional relative entropy between them is \begin{align} D(p\parallel q)& =\sum_a p(a)\sum_b p(b\mid a)\log\frac{p(b\mid a)}{q(b\mid a)}\\ & =\...
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0answers
37 views

Example of a sequence of logarithmically convex functions on $\mathbb{R}$ and for all $n\in\mathbb{N}$ in the spirit of one evoked in an article

To ask this question I was inspired in some words, if I understand well, from the authors of a preprint on arXiv in section 4.1, that I believe that is [1], to ask next question. We consider the ...
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1answer
89 views

A property of convex cones in Euclidean spaces

EDIT: Let $K$ be a closed convex cone in a Euclidean space of finite dimension. Assume it is non-zero and not the whole space. Does there exist a non-zero point $x\in K$ such that $$(x,y)\geq 0 ...
2
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1answer
130 views

Monotone function which is separately convex but not convex

I am looking for a function $f\,:\,\mathbb{R}^n\to\mathbb{R}$ which is separately convex and increasing but not convex. That is to say, the function is convex and increasing in each coordinate while ...
2
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1answer
98 views

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 \...
1
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1answer
53 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
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1answer
181 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
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1answer
271 views

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) + \...
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0answers
31 views

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 ...
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0answers
69 views

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
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0answers
76 views

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
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1answer
104 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
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1answer
257 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
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1answer
218 views

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 $\...
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0answers
49 views

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
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1answer
86 views

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
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1answer
157 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 ...

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