Questions tagged [convexity]
For questions involving the concept of convexity
189 questions with no upvoted or accepted answers
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Injectivity of a convex combination of squares $A^*A$ in $\ell^\infty$
Consider two operators $A,B: \ell^p \to \ell^p$ ([defined and]bounded for all $p \in [1,\infty]$) as well as their adjoints $A^*,B^*:\ell^p \to \ell^p$. Assume $A^*A$ and $B^*B$ have trivial kernel ...
- 4,432
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97
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Projection onto level set of convex functional
Fix a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and let $F:L^2_{\mathbb{P}}(\mathcal{F})\rightarrow (-\infty,\infty]$ be bounded-blow, convex, lower semi-continuous, and not identically ...
- 5,405
3
votes
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54
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Independence-like property of convex combinations in a vector space
Consider the following property of a set of vectors $S\subset V$, where $V$ is a real vector space:
$$
\sum_{i=1}^m w_ix_i
= \sum_{i=1}^m u_iy_i,\quad
x_i,y_i\in S,\quad
0\le w_i,u_i\le 1, \quad
\...
- 31
3
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0
answers
151
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Concavity of distance to the boundary of Riemannian manifold
Let $(M^n,g)$ be a smooth Riemannian manifold with non-empty boundary $\partial M$. Assume (for simplicity) that $M$ is compact. Let $M$ be locally geodesically convex, i.e. any shortest path in $M$ ...
- 21.8k
3
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48
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A complete metric space with some convex-type property
Let $(X,d)$ be a complete metric space with this property:
for each $x∈X$, $r>0$ and $y∈X$ with $d(x,y)<r$, there exists $z∈X$ such that $d(x,y)+d(y,z)=d(x,z)=r$.
I want to know if the family ...
- 179
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0
answers
95
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Effective radius of section of a convex set compared to that of the convex itself
The effective radius $er(A)$ of a $n$-solid $A$, is defined by Schramm (see the question by Gil Kalai
Volumes of Sets of Constant Width in High Dimensions)
to be the radius of the $n$-ball that has ...
- 469
3
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112
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Sufficient condition for convex conjugate to be second-order differentiable
Let $f:\mathbb{R}^n\to (-\infty,\infty]$ be a convex lower-semicontinuous function, we then define its conjugate by
$$
f^*(y)=\sup_{x\in \mathbb{R}^n}\{x^Ty-f(x)\}.
$$
Then there exist well-known ...
- 503
3
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116
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convex approximation for a non convex function
Consider the function
$f\left( {{x_1},...,{x_M},{y_1},...,{y_N}} \right) = \left( {\sum\limits_{j = 1}^M {{\alpha _j}{x_j}} } \right)\left( {{e^{ - \sum\limits_{i = 1}^N {{\beta _i}{y_i}} }}} \right)$...
- 275
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1
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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 ...
- 13.2k
3
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407
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Geodesically convex neighborhood in Finsler manifolds
It is well known that every point of a Riemannian manifold $(M,g)$ possesses a fundamental system $\{U_n\}_{n\in\mathbb N}$ of geodesically convex neighborhoods. This means that every pair of points ...
3
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207
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proving quasi convexity of multivariable function
Given
an arbitrary $(N \times N)$ square matrix ${\bf X}$
a positive definite $(M\times M)$ matrix ${\bf T}$
a $(Q\times MN), Q< MN$ matrix ${\bf Z}$ consisting of only 1s and 0s where there is
...
3
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141
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Covering fat objects with fat objects
The family of rectangles has the cover property, i.e.:
For every $R\geq 1$, $k\geq 1$: every rectangle with aspect ratio $kR$ can be exactly covered by $\lceil k\rceil$ (possibly overlapping) ...
- 3,549
3
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answers
159
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Strictly Convex Smoothing of a function defined on an affine manifold
A function defined on an interval is strongly convex if it has positive definite second derivative. A function on an affine manifold is strongly convex
if its restriction to each line is. An affine ...
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3
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Are there pathological examples of log-concave measures that admit no shifts?
Does there exist a random vector $X$ in, say, the space $\mathbb{R}^\infty$ of sequences that has the following properties?
The distribution of $X$ is log-concave, i.e. for every $n$ the joint ...
- 4,010
3
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188
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Covering points with a convex hull
Consider a set of $n$ points $x_1,\ldots,x_n \in \mathbb{R}^d$, for some $n \gg d$. Suppose $\{x_1,\ldots,x_n\} \subset C \subset \mathbb{R}^d$. Say that a set of points $y_1,\ldots,y_m \in C$ covers $...
- 794
3
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187
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A subclass of log-concave functions satifying a sum inequality
Suppose $f:[0,\infty)\to[0,\infty)$ is continuous and for all positive integers $m$ and real $x\geq0$, $\alpha,\beta>0$:
$$
\sum\limits_{k+n=m}[f(x+\alpha+k)f(x+\beta+n)-f(x+k)f(x+\alpha+\beta+n)]\...
- 91
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191
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Characterizing curves that bound strictly convex regions
Consider a closed curve on the plane so that if we perform any translation and dilation on it, the resulting curve intersects the original curve at most twice. Does this property characterize the ...
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Why cannot we adapt Barvinok type counting techniques to general convex integer programs?
Decision problems in Integer Linear Programming have Lenstra type algorithms (https://www.math.leidenuniv.nl/~lenstrahw/PUBLICATIONS/1983i/art.pdf) have been generalized to convex integer program ...
- 13.9k
2
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58
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An s-convex function lying between two convex functions
Let $f: \mathbb R_{+} \to \mathbb R_{+}$ be an $s$-function in the second sense, i.e.,
$$ f(\lambda x +(1-\lambda)y) \leq \lambda^s f(x) +(1-\lambda)^s f(y)$$ for every $\lambda \in (0,1)$. Assume ...
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107
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Reference request: books on convex analysis / geometry
I am interested in convex geometry and analysis, especially in its connections with high dimensional probability theory.
I was reading the book by Pisier, The volume of convex bodies and Banach space ...
- 410
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answers
164
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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 ...
- 873
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82
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A convex version of the small uncountable cardinal $\mathfrak b$
Let us recall that $\mathfrak b$ is the smallest cardinality of a subset of $\omega^\omega$, which cannot be covered by countably many compact subsets of $\omega^\omega$.
The definition of $\mathfrak ...
- 41.8k
2
votes
0
answers
61
views
Does absolute retract imply convex structure?
In the theory of selection, it is known that any compact absolute retract (AR) carries a convexity structure defined by E. Michael. It is also known that a convex structure
developed by Van de Vel ...
- 2,083
2
votes
0
answers
108
views
Characterization of inverse limits of finite-dimensional convex cones
Consider a countable inverse system $C_1\substack{f_1 \\ \leftarrow} C_2 \substack{f_2 \\ \leftarrow} C_3 \substack{f_3 \\ \leftarrow} \ldots$ where the $C_i$ are finite-dimensional convex cones of ...
- 21
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155
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Inscribed square and convexity
Let $b(X)$ be the boundary of any $X$ subset of the plane.
Does there exist $A,B$ convex compact sets of the plane, such that $C:=A\setminus B$ is simply connected and not empty, and such that ...
- 469
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answers
76
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Biconjugate of a quasiconvex lower semi-continuous function
Let $f:\mathbb{R}^d \to [0,\infty]$ be a quasiconvex lower semi-continuous function whose effective domain $C:=\{x \in \mathbb{R}^d:f(x) < \infty\}$ is nonempty and bounded (and convex since $f$ is ...
- 139
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102
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How to prove/disprove this surface integral is convex?
This question is related to the following:
Convexity of volume in terms of a deformation - the context is summarized below for clarity.
In the setting of convex optimization, I am looking for a convex ...
- 51
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answers
138
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Convexity of volume in terms of a deformation
In the context of convex optimization and mechanics, I am interested in the convexity of the potential energy $U$ of a pressure acting over some volume $V$ enclosed by a surface. Here pressure can be ...
- 51
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51
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Conjugate of composition in Bochner spaces
Let $H$ be a separable Hilbert space (of non-zero dimension), let $(\Omega,\Sigma,\mu)$ be a finite measure space, and let $L^2(\mu;H)$ be the Bochner-space $\mu$-integrable $H$-valued functions. ...
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101
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Sparse signal recovery (nonlinear case)
Let $K \subset \mathbb{R}^n$, it may be that $K$ is "very thin" (e.g. $K$ is a $k$-dimensional affine subset of $\mathbb{R}^n$, with $k \ll n$). I'm interested in the case where $K$ is ...
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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
$$
\begin{cases}
∆ u=1\quad\text{in $\Omega$,}
\\
\frac{\partial u}{\partial \nu}=\frac{\vert \Omega\vert_n}...
- 16.2k
2
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0
answers
60
views
Convexity on large scales
Has the following concept ever been studied/have a standard name?
Let $f:\mathbb{R}\to \mathbb{R}$ be continuous. We say $f$ is (mid-point) convex on large scales provided
$$f\left( \frac{x+y}{2}\...
- 2,478
2
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answers
59
views
Weak convexity in graphs
I asked the following question in MathStackExchange, but I did not get any answer, and I think that this might be the appropriate venue for the question.
As we know, a finite undirected graph ...
- 123
2
votes
0
answers
112
views
Star-convex curve and Fourier series
Let x(t) be a periodic function on [0, 2$\pi$].
I am interested in finding criteria in terms of the Fourier coefficients of x(t), such that the parametric curve $\left\{ x\left( t\right) ,\dot{x}\left(...
- 21
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votes
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answers
58
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Convex solutions of linear hyperbolic PDEs in a planar domain
Consider a linear homogeneous 2nd-order PDE in a convex planar domain $\Omega$ :
$$a(x,y)\frac{\partial^2u}{\partial x^2}+2b(x,y)\frac{\partial^2u}{\partial x\partial y}+c(x,y)\frac{\partial^2u}{\...
- 52.3k
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351
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Orthonormal Basis for Convex Functions
Are there any orthonormal bases for strictly convex functions $f: \mathbb{R}^n\ni x \mapsto \mathbb{R},\ x\ne y\implies f\left(\alpha x+\left(1-\alpha\right)y\right) \lt \alpha f(x)+(1-\alpha)f(y) \...
- 13.2k
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lower semicontinuity of the number of extreme points
Do you know the reference for the following fact:
the number of extreme points of a compact convex
subset of a locally compact space varies lower semicontinuously when we endow the space of compact ...
- 111
2
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answers
61
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Trying to show expected wait is convex -- need to show an expression is positive
I need to show that the following expression is positive
$$ (B+1) (2 B+1) z_0^B-(B+2) (\rho +1) z_0-2 (B+1) (B-1) ((\rho +1) z_0-\rho )+(B-1) (\rho +1) > 0 $$
where $B\geq 1$ is an integer, $0<...
- 63
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385
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(Quasi) convexity of separately convex homogeneous functions
Consider a function $f:\mathbb{R}^n_{\geq 0}\rightarrow \mathbb{R}$ that is separately convex, i.e. such that $\frac{d^2f}{dx_i^2}\geq 0$ for all $i\in \{1,\dots n\}$. Assume also that $f$ is ...
- 389
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views
Is there a "last mile" criterion for a generalization of planar convex hulls to symmetric weighted graphs?
This question is motivated by an almost successful attempt to define planar convex hulls of a finite set of isolated points only via subset sums of the distances between point-pairs; the advantage of ...
- 13.2k
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answers
97
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Calculating a Combinatorial Generalization of Planar Convex Hulls
In this question I have suggested a generalization of the notion of a set of points in the Euclidean plane being in convex configuration to the set of vertices of symmetric weighted graphs via $k$-...
- 13.2k
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216
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When can sublinear growth imply concavity?
Consider a function $f(x,\lambda):\mathbb{R}^{2}_{+}\to\mathbb{R}_{+}$ that is uniformly continuous, smooth, lower bounded and convex. Let
$\qquad g(\lambda)=\inf_{x}\;f(x,\lambda)$
We know that $g(...
- 141
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votes
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answers
136
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Fixed area, largest mass -- is there a name?
Let $x\in \mathbb{R}^n$ and let $s_k(x)$ denote the sum of the $k$ largest entries of $x$. The function $s_k(x)$ is well-known to be convex and is often used in optimization, such as
https://inst....
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answers
106
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Regularity of the Minkowski functionnal of a convex
Let $K$ be a convex compact set in $\mathbb{R}^2$ with $0 \in \overset{\circ}{K}$. The Minkowski functional associated to K is:
\begin{align*}
\varphi_K(x):= \inf \left\{t>0 \; : \; tx \in K \...
- 21
2
votes
0
answers
296
views
Lower convex envelope of a function involving entropy
Suppose two discrete random variables $X$ and $Y$ defined on finite sets $\mathcal{X}$ and $\mathcal{Y}$ are given and also suppose the conditional distribution $P_{Y|X}$ (i.e, channel) is fixed. We ...
- 1,109
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497
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Given a multivariate polynomial with even degree, can we find its tightest convex polynomial 'envelop'?
To be specific, given a multivariate polynomial function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ with even degree $2d$, can we construct a convex polynomial function $g$, such that:
$\forall \mathbf{...
- 81
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0
answers
151
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Why pseudoconvexity is important in Partial differential equation theory?
I am a new researcher in mathematics and I work on convexity. Are convexity and pseudoconvexity related topics and in which respect to PDE theory ? One of the important results in PDE theory is the ...
- 308
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0
answers
51
views
Conditions under which a set of points have a low weight representation under some basis
Let $S \subset R^d$ be a convex polytope in $d$ dimensional real space. Say that $S$ has weight $w$ if there exists some basis $x_1,\ldots,x_d$ such that for every point $v \in S$, we can write $v = \...
- 794
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0
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164
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Minimally 6-connected 3D discrete lines that are convex lattice sets
There are several definitions of 3D discrete lines, e.g. http://diwww.epfl.ch/w3lsp/publications/discretegeo/nratddl.html , http://dx.doi.org/10.1007/978-3-642-19867-0_4 . However, I know of none that ...
- 567
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0
answers
366
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Are affine continuous functions on Bauer sub-simplices of the probability measures given by integration over continuous functions?
Let $X$ be a compact (non-metrizable) Hausdorff space and $\mathcal{P}(X)$ the set of Radon probability measures with
weak-$*$ topology (weak topology induced by the continuous functions).
Consider a ...
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