Questions tagged [convexity]
For questions involving the concept of convexity
627
questions
4
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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 ...
4
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1
answer
355
views
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 ...
4
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0
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130
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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:
$$\alpha(A)=\sup_{b\in\overline{\operatorname{...
4
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0
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144
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How many convex or concave subsets are contained in an arbitrary set of $n$ real numbers?
This question is closely related to this post. A set $A=\{a_1<a_2<\dots<a_n\} \subset \mathbb R$ is said to be convex if the consecutive differences are non-decreasing, i.e. if $a_{j+1} - a_j ...
4
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0
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63
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Length of curves on convex hypersurfaces
Let $\gamma\colon[a,b] \to \mathbb{R}^n$ be a smooth curve.
Let $f_i\to f$ be a sequence of convex functions on $\mathbb{R}^n$ converging uniformly on compact subsets to $f$.
Let $\hat\gamma(t):=(\...
4
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0
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63
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A standard name of a strongly extremal point of a convex set
I need to name somehow points $x$ of a bounded convex set $C$ in a Banach space $X$ such that the set $$\{x^*\in X^*:x^*(x)=\max x^*[C]\}$$ of support functionals at $x$ has non-empty interior in the ...
4
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345
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On intrinsic volumes
Let $\Gamma$ be a convex polytope in $\mathbb R^n$. The $k$-th intrinsic volume of $\Gamma$ is the number
$$
\text{v}_k(\Gamma)=\sum_{\Delta\in{\mathcal B}(\Gamma,k)}\text{vol}_k(\Delta)\psi_\Gamma(\...
4
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68
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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 ...
4
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0
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534
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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 \{...
4
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0
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235
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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
$$ \|X\|_{p_1,p_2} = \left( \sum_{j=1}^n \left( \sum_{i=1}^m |x_{ij}|...
4
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0
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322
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How to check if a manifold can be foliated by strictly convex hypersurfaces?
Let $M$ be a compact Riemannian manifold with boundary.
How can one recognize whether the manifold can be foliated by strictly convex hypersurfaces?
An exact definition is given below.
If the ...
4
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0
answers
182
views
This function looks quasiconvex, can't understand why
Suppose that $\mathbf{C}$ is a given matrix with non-negative entries in $\mathbb{R}^{m\times n}$ and $d$ is a given scalar, and let $g(\mathbf{y})$ be defined by $$g(\mathbf{y}):=\max_{\mathbf{x}\in\...
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149
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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) ( \...
4
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0
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292
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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 ...
4
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158
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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 $K_{1}+K_{2}$ ...
4
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0
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119
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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 ...
3
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3
answers
1k
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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?
3
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2
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594
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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!
3
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2
answers
284
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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:...
3
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3
answers
630
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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 ...
3
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1
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274
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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 $\...
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 ...
3
votes
1
answer
231
views
$\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 $f$, $g$:
$$\int_0^1 f(\sin(1/x)) \times g(\cos(1/x)) dx \leq \int_0^...
3
votes
1
answer
134
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Busemann-Feller lemma in hyperbolic space
The classical Busemann-Feller lemma in Euclidean space says the following.
Let $K\subset \mathbb{R}^n$ be a closed convex set.
Then
for any point $x\in \mathbb{R}^n$ there exists unique nearest point ...
3
votes
1
answer
600
views
Are polynomials with only real zeros log concave functions?
Consider a polynomial $\sum\limits_{k=0}^n a_kx^k$ with $a_k\geq 0$ and $x\geq 0$.
In this comment, Richard Stanley mentions that polynomials with only real roots are log concave functions. Can ...
3
votes
2
answers
509
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Submanifolds lying on the boundary of a convex domain
Let $M$ be a submanifold of $\mathbb R^n$. Call $M$ locally convex if locally $M$ is contained in the boundary of a convex domain of $\mathbb R^n$.
Is there any known condition that is equivalent to ...
3
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2
answers
630
views
Gradient flows: convex potential vs. contractive flow?
Take a $\mathcal C^2$ potential $V:\mathbb R^d\to \mathbb R$, and assume that it is bounded from below (say $\min V=0$ for simplicity, so that $V\geq 0$).
Consider the autonomous gradient-flow
$$
\dot ...
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 ...
3
votes
1
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546
views
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 $\mathbf{R}^{n}-0$....
3
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1
answer
497
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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 ...
3
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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 ...
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= \...
3
votes
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 ...
3
votes
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 $\...
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 ...
3
votes
1
answer
208
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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 ...
3
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1
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156
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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 ...
3
votes
1
answer
245
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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 ...
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 $K$?
3
votes
1
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314
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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 ...
3
votes
1
answer
149
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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 ...
3
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1
answer
1k
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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 ...
3
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1
answer
642
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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 ...
3
votes
1
answer
423
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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 ...
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 _{...
3
votes
1
answer
342
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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 ...
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 $\...
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\...
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 ...
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 ...