Questions tagged [convexity]
For questions involving the concept of convexity
627
questions
5
votes
2
answers
245
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Which convex subsets of a normed space are intersections of balls?
Let $(V, \lVert \cdot \rVert)$ be a normed space. For any $A \subseteq V$, let $O(A)$ be the intersection of all closed balls containing $A$, or more precisely, let $O \colon 2^V \to 2^V$ be defined ...
3
votes
0
answers
85
views
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 ...
1
vote
1
answer
366
views
Convexification of difference of convex functions
I am looking for a reference/a hint to the following problem:
We are given $f_1(x),f_2(x)$ convex functions (say, on $\mathbb R^d$) such that $f_1(x) \to\infty$ for $\|x\|\to\infty$. Also there is an $...
19
votes
4
answers
3k
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Strange result about convexity
$f \in C^2([0,1])$ with $f''$ convex and $f(0) = f'(0) = f''(0) = 0$.
Is it true that : $f''(1)+6f(1)\geq 4f'(1)$ ?
Source: AoPS
1
vote
0
answers
79
views
Maximal geodesically convex function interpolating three points on the hyperbolic plane
Crossposted on MSE: https://math.stackexchange.com/questions/4282998/maximal-geodesically-convex-function-interpolating-three-points-on-the-hyperboli
Let $M$ be a two-dimensional Hadamard manifold. ...
6
votes
0
answers
192
views
Does the ball maximize the "kissing probability" of symmetric convex bodies? [duplicate]
Given a symmetric convex body $K \subset \mathbb{R}^n$ (i.e., a bounded symmetric convex set with non-empty interior), I am interested in the following quantity
$$p_K := \Pr_{x_1, x_2 \sim K}[x_1 \in ...
1
vote
0
answers
112
views
How to prove the inequality $\ln\frac{1+e^{-y}}{1+e^{-x}}+\frac{1}{1+e^x}(y-x)\geq 0$?
I am trying to prove that $\ell(\beta) = \sum_{i=1}^n \left (-y_i \beta^{\top}x_i + \ln \left (1 + e^{\beta^{\top}x_i }\right )\right )$ is a convex function. I follow the following steps:
Let $\...
4
votes
2
answers
978
views
Convexity of $(X, y) \mapsto y^T X^{-1} y$ [closed]
Let $y \in \mathbb{R}^n$, $X \in \mathcal{S}^n_{++}(\mathbb{R})$. Why would function $ f : (X, y) \mapsto y^T X^{-1} y$ be convex?
I tried with $(X, x) + t.(Y, y)$ with no result. Also, I thought ...
2
votes
0
answers
99
views
Characterization of inverse limits of finite-dimensional convex cones
Consider a countable inverse system $C_1\substack{f_1 \\ \leftarrow} C_2 \substack{f_2 \\ \leftarrow} C_3 \substack{f_3 \\ \leftarrow} \ldots$ where the $C_i$ are finite-dimensional convex cones of ...
2
votes
1
answer
169
views
When is a continuous subadditive function (0,1]-superhomogeneous
Continuous version of this Superhomogeneity of subadditive functions
Let $f$ be a continuous function of $\geq 2$ real variables defined on a convex cone $\mathcal{C}$ in the upper half plane, with $f(...
2
votes
1
answer
97
views
Superhomogeneity of subadditive functions
Let $f$ be a function of $\geq 2$ real variables defined on a convex cone $\mathcal{C}$ in the upper half plane, with $f(0) = 0$. Suppose $f$ is subadditive, i.e. $f(x_1+y_1, \dots,
x_n+y_n) \leq f(...
5
votes
0
answers
239
views
Log-concavity of lattice-functions and convolution
I was looking at the definition of log-concavity:
A function $F:\mathbb{R}^n\rightarrow\mathbb{R}$ is said log-concave
iff $F(x)\geq 0\forall x\in\mathbb{R}^n$ and
$$F(x)^\lambda F(y)^{1-\lambda}\leq ...
1
vote
1
answer
83
views
Worst convex compact set for translational packings of $\mathbb R^d$
A (translational) packing of a convex compact subset (with non-empty interior) $\mathcal C$ of $\mathbb R^d$ is a union
of translated non-overlapping (but perhaps touching) copies
of $\mathcal C$.
The ...
0
votes
1
answer
184
views
Log-concavity of the modified Bessel function of a second kind
I was searching for some results for the log-concavity of the modified Bessel function of a second type, but I failed. Has there been any known work on this? I am not even sure if it is the modified ...
2
votes
0
answers
121
views
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 ...
4
votes
0
answers
63
views
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):=(\...
3
votes
1
answer
134
views
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 ...
1
vote
1
answer
295
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
179
views
Factorizing the doubly stochastic matrix where all entries are equal such that the factors are all convex combinations of few permutation matrices
Let $N_{n}=(1/n)_{i=1,j=1}^{n}$ be the $n\times n$-matrix where all the entries are equal.
Suppose $n>0$. Let $\delta_{n}$ be the least natural number such that $N_{n}$ can be factored as $N_{n}=A_{...
2
votes
1
answer
259
views
Can the subdifferential become unbounded at interior points?
Consider $f: \mathbb{R}^n \to \overline{\mathbb{R}}$ a lower-semicontinuous, proper, closed and convex. My question is, can the subdifferential of $\partial f$ be unbounded in the interior of $\text{...
2
votes
0
answers
71
views
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 ...
0
votes
0
answers
254
views
Convexity of a set of probability densities
Consider the space of probability densities $(P(\mathbb{R}^d), W_2)$ (probability measures on $\mathbb{R}^d$ with 2-Wasserstein distance).
How can we determine if a subset $Q$ is convex?
I know that a ...
2
votes
0
answers
102
views
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 ...
2
votes
0
answers
136
views
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 ...
3
votes
0
answers
52
views
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:
$$
\sum_{i=1}^m w_ix_i
= \sum_{i=1}^m u_iy_i,\quad
x_i,y_i\in S,\quad
0\le w_i,u_i\le 1, \quad
\...
1
vote
0
answers
826
views
The "interior" of a convex set?
Consider any compact metric space $S$ which is a convex subset of a vector space. I am trying to see if the following definition is equivalent/close to some well-known definitions.
Consider a convex ...
9
votes
1
answer
717
views
Simple-looking problem with integrals
Let $f: [0,\infty) \rightarrow \mathbb{R}$ be a continuous function such that $f(0) = 0$. Is it true
that if the integral
$$
\int_0^{\pi/2} \sin(\theta) f(\lambda \sin(\theta)) \, d\theta
$$
is zero ...
2
votes
1
answer
123
views
Positive scalar curvature on the double of a manifold (assuming mean convexity of the boundary)
This question is related to a previous one.
Let $(M^n,g)$ be a compact Riemannian manifold with boundary. Assume it has positive scalar curvature and $\partial M$ is mean convex (positive mean ...
-4
votes
1
answer
266
views
strict convexity and Lipschitz continuity [closed]
Consider a continuously differentiable function $f: \mathbb{R}^n \mapsto \mathbb{R}$. If $f$ is strictly convex, does it imply that it is not Lipschitz on $\mathbb{R}^n$?
Because if $f$ is strictly ...
1
vote
0
answers
121
views
Relation satisfied by a Gaussian random variable
I want to prove the following relation for $X\sim \mathcal{N}(0,1)$, $x\in \mathbb{R}$ and $f(x)=\mathbb{E}[\max(X,x)]$:
$$f(\frac{f(x+1)+f(x-1)}{2})\leq \frac{f(f(x)-1)+f(f(x)+1)}{2}$$
It seems that ...
6
votes
0
answers
132
views
Nearby convex set in a nearby space
Let $K$ be a convex set in a CAT(0) space $X$. Suppose $X'$ is a CAT(0) space that is very close to $X$.
Is there a convex set $K'\subset X'$ that is close to $K\subset X$?
Two spaces $X$ and $X'$ ...
4
votes
1
answer
279
views
Intrinsic definition of a cone in a normal fan
Let $P\subseteq \mathbb{R}^n$ be a full dimensional polytope. Let us assume that $P$ has a facet description with the following inequalities:
$$ \left<x,u_F\right> \geq -a_F$$
where $u_F\in \...
10
votes
0
answers
213
views
Extremal bases in finite-dimensional Banach spaces
Definition. A basis $e_1,\dots,e_n$ for a Banach space $X$ is called extremal if there exists a point $s$ in the unit sphere $S_X=\{x\in X:\|x\|=1\}$ such that for every $i\in\{1,\dots,n\}$ the ...
11
votes
1
answer
212
views
The set of boundary vectors of compact convex body has empty interior
Let $K$ be a compact convex body in the Euclidean space $\mathbb R^n$ and $\partial K$ be its topological boundary in $\mathbb R^n$.
Definition. A vector $\mathbf v\in\mathbb R^n$ is called $K$-...
67
votes
3
answers
11k
views
Nonconvexity and discretization
Edit: Here's a more down-to-earth, and somewhat weakened, but I believe still nontrivial, version of the main theorem.
Prototypical nonconvex spaces are $\ell^p$-spaces for $0<p<1$, say $\ell^p(\...
4
votes
0
answers
63
views
A standard name of a strongly extremal point of a convex set
I need to name somehow points $x$ of a bounded convex set $C$ in a Banach space $X$ such that the set $$\{x^*\in X^*:x^*(x)=\max x^*[C]\}$$ of support functionals at $x$ has non-empty interior in the ...
5
votes
1
answer
205
views
Sufficient condition for geodesic convexity/connectedness
Let $(\Sigma,g)$ be a connected smooth Riemannian manifold without boundary. By a minimizing geodesic I mean a geodesic whose length equals the distance between its endpoints. Let us consider the ...
1
vote
0
answers
184
views
Stationary distributions of convex combination of stochastic matrices
Consider two irreducible finite state Markov chains with transition matrices $A,B\in\mathbb{R}^{n\times n}$.
Let $x$ and $y$ be the unique stationary distributions of $A$ and $B$, respectively.
Now ...
2
votes
0
answers
51
views
Conjugate of composition in Bochner spaces
Let $H$ be a separable Hilbert space (of non-zero dimension), let $(\Omega,\Sigma,\mu)$ be a finite measure space, and let $L^2(\mu;H)$ be the Bochner-space $\mu$-integrable $H$-valued functions. ...
1
vote
0
answers
271
views
Decomposition of Polyhedral - An example
There is no doubt that clear examples consolidate the understanding of concepts being learnt. I am new to finding the structure and decomposition of a polyhedra. Suppose that we have the system
$$ \...
0
votes
1
answer
76
views
Planar function inequality on parallelograms
Let $f$ be a function defined on the unit square $R = [0,1]^2 \subseteq \mathbf{R}^2$ which is convex and satisfies $\frac{\partial{f}^2 }{\partial{x}\partial{y}} \leq 0$. The last condition is ...
24
votes
4
answers
2k
views
Why the sequence of Bernstein polynomials of $\sqrt x$ is increasing?
Bernstein polynomials preserves nicely several global properties of the function to be approximated: if e.g. $f:[0,1]\to\mathbb R$ is non-negative, or monotone, or convex; or if it has, say, non-...
2
votes
1
answer
178
views
Sufficient conditions for the convexity of the discrete Fourier transforms
Let $f : [0,2\pi] \to \mathbb{R}$ be some function. Then the discrete Fourier transform of $f$ when sampled at $2\pi i/N$ is then given by
$$
X_n := \sum_{i=0}^{N-1}\cos\left(\frac{2\pi n i}{N}\right)...
1
vote
1
answer
283
views
Convexity of discrete Fourier transform
Let $f : [0,2\pi] \to \mathbb{R}$ be a continuous convex function on $(0,2\pi)$ which is singular about $0$ and $2\pi$ but finite when evaluated at the boundaries. Assume also that $f$ is symmetric ...
7
votes
2
answers
731
views
Is every face exposed if all extreme points are exposed?
Let $C$ be a non-empty compact convex subset of ${\mathbb R}^d$ such that every extreme point of $C$ is an exposed point of $C$. Does it follow from this that every face of $C$ is an exposed face?
1
vote
1
answer
215
views
Existence of an asymptote for $g(x)=\frac{f(x)f'(x)+f(1)f'(1)}{f'(x)+f'(1)}-f\left(\frac{xf'(x)+f'(1)}{f'(x)+f'(1)}\right)$
Working with Slater's inequality (a companion of Jensen's inequality) I found this statement:
Let $f(x)$ be a continuous, twice differentiable function, convex or concave and non constant on $(0,\...
1
vote
1
answer
38
views
Is this relation between planar convex hulls and heaviest cliques true?
If $P$ is a set of $n$ points in the euclidean plane whose convex hull $\operatorname{CH}(P)$ has $h$ corners, and $Q\subset P$ has $m\le\lfloor\frac{h}{2}\rfloor$ points and maximal sum of pairwise ...
11
votes
0
answers
356
views
Is there yet an example of a non-negative convex polynomial that cannot be written as a sum-of-squares?
I have read that it remains an open question, whether an example can be constructed of a non-negative convex polynomial that cannot be written as a sum-of-squares. My reading includes the following ...
9
votes
1
answer
301
views
Closedness of linear image of positive L1 functions
Let $\mathcal X$ be the Banach space of $L^1$ functions on some probability space, $\mathcal Y$ be some other Banach space, $T:\mathcal X\to \mathcal Y$ be some surjective continuous linear map, $\...
2
votes
0
answers
98
views
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 ...