Questions tagged [convexity]

For questions involving the concept of convexity

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Generalized widths and reverse Urysohn inequalities

This question is inspired by the discussion in MO questions "Local minimum from directional derivatives in the space of convex bodies" and "Bodies of constant width?" about generalized notions of ...
Yoav Kallus's user avatar
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Efficiently sampling points from an integer lattice.

Let $\mathcal{L}$ = {x$\in$ $N^n$ : ||x||$_1$ $\leq$ m} denote the set of integer points in the positive orthant of the $\ell_1$ ball of radius $m$, where $m < n$. For each $x \in \mathcal{L}$, let ...
Wilson's user avatar
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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{...
Motaka's user avatar
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How many convex or concave subsets are contained in an arbitrary set of $n$ real numbers?

This question is closely related to this post. A set $A=\{a_1<a_2<\dots<a_n\} \subset \mathbb R$ is said to be convex if the consecutive differences are non-decreasing, i.e. if $a_{j+1} - a_j ...
Oliver Roche-Newton's user avatar
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Length of curves on convex hypersurfaces

Let $\gamma\colon[a,b] \to \mathbb{R}^n$ be a smooth curve. Let $f_i\to f$ be a sequence of convex functions on $\mathbb{R}^n$ converging uniformly on compact subsets to $f$. Let $\hat\gamma(t):=(\...
asv's user avatar
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A standard name of a strongly extremal point of a convex set

I need to name somehow points $x$ of a bounded convex set $C$ in a Banach space $X$ such that the set $$\{x^*\in X^*:x^*(x)=\max x^*[C]\}$$ of support functionals at $x$ has non-empty interior in the ...
Taras Banakh's user avatar
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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(\...
James Silipo's user avatar
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"Singularly convex" cones of matrices

The ambient space if ${\bf M}_n({\mathbb R})$. Let us begin with facts. 1- The cone of positive semi-definite symmetric matrices is convex. 2- It is a little subtler that the cone $K^+$ of matrices ...
Denis Serre's user avatar
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Convexity of the electrostatic energy on a Riemann surface

Let $M$ be a compact Riemann surface. Let $\Lambda$ be a differentiable real $2$-form of integral one. Let $G$ be the Green function associated to $\Lambda$, i.e. $G: M \times M \to \mathbb R \cup \{...
D.E.G.Z.'s user avatar
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On the modulus of convexity of mixed-norm $\ell_{p_1,p_2}$ spaces

Let $\ell_{p_1,p_2}=(\mathbb{R}^{m\times n},\|\cdot\|_{p_1,p_2})$ be the space of $m\times n$ matrices endowed with the mixed-norm $$ \|X\|_{p_1,p_2} = \left( \sum_{j=1}^n \left( \sum_{i=1}^m |x_{ij}|...
Cristóbal Guzmán's user avatar
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How to check if a manifold can be foliated by strictly convex hypersurfaces?

Let $M$ be a compact Riemannian manifold with boundary. How can one recognize whether the manifold can be foliated by strictly convex hypersurfaces? An exact definition is given below. If the ...
Joonas Ilmavirta's user avatar
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This function looks quasiconvex, can't understand why

Suppose that $\mathbf{C}$ is a given matrix with non-negative entries in $\mathbb{R}^{m\times n}$ and $d$ is a given scalar, and let $g(\mathbf{y})$ be defined by $$g(\mathbf{y}):=\max_{\mathbf{x}\in\...
Richard Senn's user avatar
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Prove a complicated function (in epidemic spreading search) to be convex

When analyse epidemic spreading, I came across to prove that a complicated function $f(x)$ is convex when $0 \leq x \leq 1$. \begin{equation} f(x)=\frac{b_1g'(x) f_1(x)^{n-2}+g'(1) \gamma}{g'(1) ( \...
Changwang Zhang's user avatar
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When is the sum of a weak-$*$ closed convex cone and a subspace also weak-$*$ closed?

Let $X$ be a Banach space. Suppose $C \subset X^*$ is a convex cone and $V \subset X^*$ is a subspace, and suppose both $C$ and $V$ are closed in the weak-$*$ topology. Are there any general ...
Evan DeCorte's user avatar
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Hilbert metric of a sum of cones

Suppose that $K_{1}$ and $K_{2}$ are pointed closed cones in a finite-dimensional space $V$ whose Hilbert metrics $d_{1},d_{2}$ are known. Is there a way to express the Hilbert metric of $K_{1}+K_{2}$ ...
Felix Goldberg's user avatar
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119 views

Symmetry of Sundry Planar Convex Sets of Constant Width & Minimal Area

In a much broader paper in “Optimization Methods & Software 27,6 (2012) pp1073-1099” Bayen & Henrion consider planar compact, convex sets with support functions which are finite Fourier ...
Ian Calvert's user avatar
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3 answers
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Radstrom cancellation only for two convex sets?

I've seen this statement of Radstrom cancellation: if $A +C \subset B+C$ where $A,B$ are convex, $B$ is closed, and $C$ is bounded, then $A \subset B.$ Is it essential that $A$ be convex?
AatG's user avatar
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If $K$ and $L$ are compact convex sets with smooth boundary, does their union have piecewise-smooth boundary?

Clarification: by "piecewise", I mean a finite number of pieces. I'm sure this must be true, but my search for a citation was in vain (although I did learn the new term "polyconvex"). Thanks!
Ryan O'Donnell's user avatar
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2 answers
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Does the automorphism group of a cone determine the cone?

A cone is a $R_+$-module. That is, a cone is an abelian monoid that is closed under nonnegative real scalar multiplication. An automorphism of a cone is a bijective $R_+$-linear map. That is a map $f:...
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630 views

Inequalities for uniformly convex normed spaces

When reading "Chebyshev centers and uniform convexity" by Dan Amir I encountered the following result which is apparently "known and easy to prove". I'm sure it is, but I can't find a proof and am ...
David R. MacIver's user avatar
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1 answer
274 views

Signed measure that is positive over convex sets

I have a signed measure $\mu$ on a convex subset $C\subset \mathbb{R}^n$, and I want to prove that $\mu$ is a probability measure, most importantly that it is positive everywhere. I do know that $\...
Henrique de Oliveira's user avatar
3 votes
2 answers
180 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 ...
Aimar's user avatar
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$\int_0^1 f(\sin(1/x)) \times g(\cos(1/x)) dx \leq \int_0^1 f(\sin(1/x)) dx \times \int_0^1 g(\cos(1/x))dx? $

I have noticed experimentally that the following question has a positive answer. Is it true that for all even and convex functions $f$, $g$: $$\int_0^1 f(\sin(1/x)) \times g(\cos(1/x)) dx \leq \int_0^...
Dattier's user avatar
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3 votes
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Busemann-Feller lemma in hyperbolic space

The classical Busemann-Feller lemma in Euclidean space says the following. Let $K\subset \mathbb{R}^n$ be a closed convex set. Then for any point $x\in \mathbb{R}^n$ there exists unique nearest point ...
asv's user avatar
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Are polynomials with only real zeros log concave functions?

Consider a polynomial $\sum\limits_{k=0}^n a_kx^k$ with $a_k\geq 0$ and $x\geq 0$. In this comment, Richard Stanley mentions that polynomials with only real roots are log concave functions. Can ...
user_lambda's user avatar
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Submanifolds lying on the boundary of a convex domain

Let $M$ be a submanifold of $\mathbb R^n$. Call $M$ locally convex if locally $M$ is contained in the boundary of a convex domain of $\mathbb R^n$. Is there any known condition that is equivalent to ...
AndreA's user avatar
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2 answers
630 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 ...
leo monsaingeon's user avatar
3 votes
1 answer
171 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 ...
Anupam's user avatar
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1 answer
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When are cones of matrices "generated" by vectors?

The cone $P_{n}$ of positive semidefinite matrices of order $n$ can be represented in this form: $P_{n}=\{A|\forall x\geq 0: \langle A,xx^{T}>0 \rangle \}$ with $x$ running over $\mathbf{R}^{n}-0$....
Felix Goldberg's user avatar
3 votes
1 answer
497 views

Example in Guillemin-Sternberg's Convexity Paper

At the end of the introduction of Guillemin-Sternberg's 1982 Inventiones paper "Convexity Properties of the Moment Mapping", there is a nice picture of a non-Abelian moment polytope attributed to ...
Michael's user avatar
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3 votes
2 answers
255 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 ...
Iosif Pinelis's user avatar
3 votes
2 answers
221 views

Hausdorff dimension of the non-differentiability set a convex function

Let $X \subset \mathbb R^d$ be open, $f : X \to \mathbb R$ and $$ E := \{x \in X : f \text{ is not Fréchet differentiable at }x\}. $$ Then we have the following result which is Theorem: If $X= \...
Akira's user avatar
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1 answer
241 views

Sufficient conditions for weak majorization

Suppose $x_1\ge x_2\ge \cdots \ge x_n\ge 0$ and $y_1\ge y_2\ge\cdots\ge y_n\ge0$ be reals such that for any positive integer $p$, $$ \sum_{i=1}^n x_i^p \geq \sum_{i=1}^n y_i^p. $$ Question: Is ...
MERTON's user avatar
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2 answers
267 views

Eigenspace of convex combination of two idempotent matrices

Let $H_1,H_2\in\mathbb{Q}^{n\times n}$ be idempotent and symmetric matrices. For any $0<\mu<\frac{1}{2}$, consider the matrix $$H_\mu:=\mu H_1+(1-\mu)H_2.$$ I'm looking for a description of $\...
Tobias Windisch's user avatar
3 votes
1 answer
257 views

Gradient estimate of convex functions

Consider a special type of convex function $g(\cdot):\mathbb{R}^d \to \mathbb{R}_+\cup\{+\infty\}$ such that $g(x)=+\infty$ as $|x|\to \infty$. Then $g$ is differentiable almost everywhere within its ...
Roy Han's user avatar
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3 votes
1 answer
208 views

Quantitative stability: Hausdorff distance between subdifferentials

Suppose I have two convex functions $f$ and $g$ mapping $\mathbb{R}^{n}\to\mathbb{R}$ (so they are 'more' than proper). Suppose $\|f-g\|_{\infty}<\epsilon$, the sup-norm. In particular, the ...
Pallen's user avatar
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1 answer
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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 ...
elwyn's user avatar
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3 votes
1 answer
245 views

Polygon of convex arcs

Convex polygons in the plane $R^2$ arise in linear programming where the constraints are linear. The objective linear function attains its maximum at a vertex of the feasible region(if exists). Assume ...
Semsem's user avatar
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3 votes
1 answer
198 views

Convexity of the matrix mapping $X^{-2}$

Let $X$ be a positive semidefinite matrix. Is the mapping $X\to X^{-2}$ convex? Update: or is $Tr[X^{-2} K]$ convex for PSD $X$ and $K$?
Soheil Feizi's user avatar
3 votes
1 answer
314 views

Characterization of convex functions

Let $\Omega$ be a convex open subset of $\mathbb R^n$ and let $f:\Omega\rightarrow \mathbb R$ be a convex function. Since $f$ is continuous, it can be considered as a distribution on $\Omega$ and then ...
Bazin's user avatar
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3 votes
1 answer
149 views

The center of a minimal convex superbody

Is the following true? CONJECTURE: $\,$ Let $\ B\ C\subseteq\mathbb R^n\ $ be convex bodies in $\mathbb R^n$ such that $\ C\ $ is centrally symmetric, $\ B\subseteq C,\ $ and $\ t\!\cdot\! B\ $ cannot ...
Włodzimierz Holsztyński's user avatar
3 votes
1 answer
1k views

Is this function of a matrix convex?

Let $\mathcal{N}_{n}$ be the set of symmetric nonnegative irreducible matrices. For a matrix $A \in \mathcal{N}_{n}$ let $v^{A}$ be its Perron vector, normalized so that $||v^{A}||_{2}=1$. Define the ...
Felix Goldberg's user avatar
3 votes
1 answer
642 views

How to examine the convexity of a complex function numerically?

I have a function which does not have a closed form . Large numerical effort should be done to evaluate the function for even a single point. How can I examine the convexity of my function over the ...
behrad mahboobi's user avatar
3 votes
1 answer
423 views

Convexity in $\{0,1\}^n$

how is convexity defined in a subset $A \subset \{0,1\}^n$? furthermore, is there any extention of the Brunn-Minkowski inequality for subsets of $\{0,1\}^n$? thanks. Edit (previously posted as an ...
ak47's user avatar
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3 votes
1 answer
693 views

Tverberg partitions with less than (r-1)(d+1)+1 points

The Tverberg Theorem states the following: Let $x_1,x_2,\dots, x_m$ be points in $R^d$ with $m \ge (r-1)(d+1)+1$. Then there is a partition $S_1,S_2,\dots, S_r$ of $\{1,2,\dots,m\}$ such that $\cap _{...
Arnau's user avatar
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3 votes
1 answer
342 views

A differential inclusion relating to the slope of a convex function

This question is concerned with a possible lemma which would be very useful in one of my current research projects, but which I am currently unable to prove. The project as a whole relates to the ...
Ian Morris's user avatar
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3 votes
1 answer
442 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 $\...
John Sung's user avatar
  • 111
3 votes
1 answer
431 views

VC dimension under projection

Let $C$ be a family of convex sets in $\mathbb{R}^d$ and assume further that $C$ is closed under translation: for all $A\in C$ and $x\in\mathbb{R}^d$, we have $A+x\in C$. Let $P:\mathbb{R}^d\to\...
Aryeh Kontorovich's user avatar
3 votes
1 answer
316 views

Positive semi-definite in the limit

Consider the $n\times n$ matrix $F$ defined by the following expression $$ F=A-\varepsilon B $$ where $A$ is a constant matrix such that $a_{ij}=a>0$ for all $i,j$ and where $B$ is a symmetric ...
user_lambda's user avatar
3 votes
1 answer
245 views

Map from a convex polygon that increases distance

At the risk of asking an extremely stupid question, suppose that $P\subset\mathbb{R}^2$ is a convex polygon with area $1$ that contains the origin, and let $r$ denote the farthest distance between the ...
Reid Evans's user avatar

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