Questions tagged [convexity]
For questions involving the concept of convexity
625
questions
2
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1
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69
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Common boundary point of convex bodies
Let $K_1, \ldots, K_n \subset \mathbb{R}^n$ be some convex bodies (i.e., compact with nonempty interior) such that for each subset $I \subset \{1,\ldots,n\}$ the set
$$\left( \bigcap_{i \in I} K_i \...
8
votes
1
answer
272
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Almost convex combinations in $\mathbb R^n$
Working on some problems in the $C_p$-theory I discovered the following simple but amazing
Fact. For any subset $A\subset \mathbb R^n$, non-zero vector $a\in \bar A\subset\mathbb R^n$ and $\...
3
votes
1
answer
87
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Separation of Infinite-Dimensional Salient Convex Cones
Let X be the set of all summable sequences of reals endowed with the $l^1$ norm.
That is, two elements of x are
$a=(a_1,a_2,....)$ and
$b=(b_1,b_2,...)$ and $d(a,b) = \sum_n |a_n-b_n|$.
In this set ...
1
vote
1
answer
294
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Nonlinear low-rank approximation - corrected
I would like to state that this is related to a past question of mine which contained errors and now appears in the corrected form, with the erroneous one deleted and closed.
In my research of linear ...
1
vote
1
answer
103
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Inequality satisfied for $t=1/2$ and Measurability implies Log-Concavity
Say $f:\mathbb{R}\to(0,\infty)$ is measurable, and that
$$\forall a,b\in\mathbb{R}~~a<b\implies\log f(a)+\log f(b)\leq2\log f(\frac{a+b}{2}).$$
Why must $f$ be log-concave? (That is, why must
$$\...
6
votes
3
answers
810
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Mixtures of log-convex functions are log-convex: a reference
A referee of a submitted paper requested details on the statement that $\int_0^a e^{-tx^2}\,dx$ is log-convex in real $t$, for each $a>0$. While there are a number of ways to prove this statement, ...
4
votes
0
answers
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 ...
8
votes
2
answers
675
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Faces of the intersection of convex sets
Let $V$ be a normed real vector space and let $K_1, K_2\subseteq V$ be closed convex subsets such that the intersection $K_1\cap K_2$ is non-empty. Assume that $F_1$ is a face of $K_1$ and $F_2$ is a ...
2
votes
0
answers
316
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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) \...
10
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0
answers
252
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Plank invariant measures on convex bodies
Let $K\subset R^2$ be a convex body, i.e., a compact convex set with interior points. A plank $P$ is the region between a pair of parallel lines in $R^2$. Let us say that $P$ intersects $K$ properly ...
37
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0
answers
1k
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Converse of the Archimedean property of the sphere
In his remarkable book On the Sphere and Cylinder, where he came tantalizingly close to discovering calculus, Archimedes showed that the area of the portion of the sphere contained between a pair of ...
2
votes
0
answers
81
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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 ...
7
votes
3
answers
418
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Does the following statement imply convexity?
I apologize if this is a simple question or if this is not the right forum for it. Some background: the subadditivity of Shannon's entropy is credited to the concavity of $-x\log(x)$. So this got me ...
24
votes
3
answers
1k
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Average measure of intersection of a convex region with its translate
Let $\lambda$ denote the Lebesgue-measure on $\mathbb{R}^n$, and let $C\subset\mathbb{R}^n$ be a convex region.
My question is about
$$f(C):=\int_{C} \lambda(C \cap (x + C) ) \mathrm{d} x.$$
How ...
7
votes
1
answer
135
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Monotonicity of canonical ellipsoids
Let $\mathcal{C}$ be the set of compact convex centrally symmetric sets in $\mathbb{R}^d$, and let $\mathcal{E} \subset \mathcal{C}$ be the set of ellipsoids centered at the origin.
I'm looking for a ...
1
vote
1
answer
71
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Properties of Relative Entropies
I'm looking at a paper by Arnold et al. (CPDE, 2001), in which they make use of convex functions in the context of relative entropies. There, they assume that on $(0,\infty)$, their entropy function ...
16
votes
4
answers
1k
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Proof of complete monotonicity of a binomial function
By plotting the function and its derivatives, one can easily be convinced that the function
$$f(x):=\log\binom{x}{p x}=\log\Gamma(x+1)-\log\Gamma(px+1)-\log\Gamma((1-p)x+1),$$ defined for $x>0$ and ...
1
vote
0
answers
88
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On convex quadratic programming clarification
We know convex quadratic programming is in $P$.
Is it also in $P$ if the function of interest is only convex in the domain of interest?
1
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0
answers
146
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Coordinate descent conditions
The following is quoted from "Bertsekas, D. P. (1999). Nonlinear programming (p. 794). Belmont: Athena scientific".
Convergence of Coordinate Descent: Suppose a function $f$ is continuously ...
1
vote
1
answer
582
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Extreme Points of a set of distributions with moment and/or support constraint
Let $X$ be a random variable with the distribution $F$ (cdf).
What are the extreme points of the sets of the form:
\begin{align}
P_1&=\left\{ F: \int |x|^k dF\le c \right\},\\
P_2&=\left\{ F:...
7
votes
0
answers
869
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Geometry of level sets of a convex function
EDIT: Let $\Omega\subset \mathbb{R}^n$ be an open subset. Let $f\colon \Omega\to\mathbb{R}$ be a function such that for some $\lambda$ the function $f(x)+\lambda |x|^2$ is convex. Assume that the ...
3
votes
1
answer
604
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Extreme points of set of probability measures $\mathcal{P}= \{F: \int_{\mathbb{R}} |x|^k dF(x)=c \}$
I am interested in finding the extreme points of the following set of distributions
\begin{align}
\mathcal{P}= \left\{F: \int_{\mathbb{R}} |x|^k dF(x)=c \right\}
\end{align}
where $k,c>0$.
I ...
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\...
2
votes
1
answer
211
views
Checking concavity of a highly non linear function
I have a highly non linear profit function which depends on four independent variables (decision variables) E,W,T and p. I want to check concavity of profit function with respect to these four ...
6
votes
1
answer
231
views
About the existence of a convergent sequence
Let $(A_n)$ be a set sequence in a Banach space wheresuch that $A_n$ is nonempty, closed and convex for every $n=1,2\dots$. Assume that $\displaystyle\lim_{n,m\to \infty} d(A_n,A_m)=0$ where d is the ...
2
votes
0
answers
59
views
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<...
5
votes
1
answer
519
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When minimum of two supporting functionals of convex bodies is convex?
For a convex compact set $K\subset \mathbb{R}^n$ let us denote by $h_K$ its supporting functional
$$h_K(\xi):=\sup_{x\in K}\langle\xi,x\rangle.$$
Thus $h_K\colon \mathbb{R}^n\to \mathbb{R}$ is a ...
3
votes
1
answer
313
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 ...
1
vote
1
answer
2k
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minimum of convex function in different variables [closed]
Let $g_1,g_2$ be convex functions defined over $[0,1]$, and let $f:[0,1]^2 \rightarrow\mathbb R$ such that
$$f(x,y)=\min(g_1(x),g_2(y)). $$
I wish to know whether $f$ is convex. I do suspect that $f$ ...
8
votes
0
answers
192
views
Concavity of product and ratio of sums
Apologies if this question is not appropriate for MathOverflow. I have asked at Math.StackExchange without success.
Consider the function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ defined as
$$
f(x)=\...
3
votes
1
answer
596
views
Are polynomials with only real zeros log concave functions?
Consider a polynomial $\sum\limits_{k=0}^n a_kx^k$ with $a_k\geq 0$ and $x\geq 0$.
In this comment, Richard Stanley mentions that polynomials with only real roots are log concave functions. Can ...
16
votes
1
answer
481
views
Bull's-eye Riemann sum
Let $f:[a,b] \to \mathbb{R}^2$ be a continuous curve on the plane.
Question: Are there numbers $a \leq x \leq c \leq y \le b$ such that $$(c-a)f(x)+(b-c)f(y) = \int_a^b f(t) \, dt \ ?$$
In other ...
9
votes
2
answers
479
views
Comparing the growth of $f\circ g$ and $g\circ f$
I asked this Question on Math.StackExchange without success. Then I learned, that this might be the better place to ask. So, sorry for crossposting. I would agree on deleting my old question.
Let $\...
5
votes
2
answers
311
views
Convex hull with genus information
Are there convexity generalizations that admit genus information?
For example in genus $1$ is there a way to think of this polyhedron as convex while this polyhedron as non-convex? Any two points can ...
0
votes
0
answers
95
views
Is there any concise sufficient condition for the dual space to have Kadec property?
A normed space $E$ has a
Kadec property if the norm- and weak topologies coincide on the unit sphere of $E$.
Kadec-Klee property if any sequence on the unit sphere, that is weakly convergent is also ...
2
votes
1
answer
223
views
Conditions for a monotonic integral average
I am looking for conditions that ensure that an integral average of a function from $\mathbb R^n$ to $\mathbb R$ is a monotonic function of the averaging set.
To be more specific, let me start with ...
1
vote
1
answer
214
views
A generalization of strict convexity
Consider the following properties of a Banach space:
the intersection of any support hyperplane with the unit sphere is
(S) a singleton (this is the strict convexity);
(SF) finite-dimensional set;...
1
vote
0
answers
175
views
Continuity of a convex function on a vector bundle
Consider the rank-${n \choose m}$ vector bundle $\pi\colon E:=\bigwedge^m(TN)\to N$ over a smooth Finsler manifold $N$ and equip each fibre $E_q := \pi^{-1}(q)$ with a norm that depends smoothly on $q\...
18
votes
1
answer
2k
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How bad can the second derivative of a convex function be?
One can easily construct an example of a measurable function $f:(a,b)\to \mathbb{R}$ which satisfies the following property:
$$\label{p}\tag{P}
f\notin L^1(I),\ \mbox{for each interval}\ I\subset (a,...
2
votes
2
answers
120
views
Convexity inequality
Let $E$ be a subset of $\mathbb{R}^n$ such that $\mathbb{R}^n \setminus E$ is convex.
Let $x,y$ be in $\mathbb{R}^n$. Is it true that for $t\in [0,1]$, we have: $$d(tx+(1-t)y,E) \geq td(x,E) - (1-t)d(...
3
votes
1
answer
313
views
Characterization of convex functions
Let $\Omega$ be a convex open subset of $\mathbb R^n$ and let $f:\Omega\rightarrow \mathbb R$ be a convex function. Since $f$ is continuous, it can be considered as a distribution on $\Omega$ and then ...
38
votes
2
answers
2k
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How to make a sandwich from just one piece of bread?
I don't know how to go about such questions. It's not exactly my area, so maybe it is stupid, but curiosity is winning.
So I have a piece of bread $P$ of a really non-regular shape (let's make it ...
4
votes
1
answer
254
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Generalization of Radon's theorem
A Cat(0) metric space $(X,d)$ of constant and finite local dimension is approximately flat if there exists a dense subset $U\subset X$ such that every $x\in U$ has a flat neighborhood (i.e. isometric ...
1
vote
1
answer
70
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Every open convex-valued multimap has global sections?
Let $X$ be a compact Polish space and $Y$ be a separable real Banach space. Assume $U \subseteq X \times Y$ is open, bounded in $Y$-norm, and s.t. for any $x \in X$, $\{y \in Y \mid (x,y) \in U\}$ is ...
44
votes
7
answers
3k
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The missing link: an inequality
I've been working on a project and proved a few relevant results, but got stuck on one tricky problem:
Conjecture. If $2\leq n\in\mathbb{N}$ and $0<x<1$ is a real number, then
$$F_n(x)=\...
0
votes
0
answers
80
views
Comparison of two functions
Given a function $f$ from $R^2$ to $R$ satisfying tha following:
$1)$ $f$ is a convex function which vanishes on $(0,1)$ and on $(1,0).$
$2)$ $f$ is a decreasing function on $x$ and on $y$ and $f$...
1
vote
1
answer
353
views
Question on Jensen's inequality
Let $(X,Y)$ be a martingale on $\mathbb R$ and $\psi:\mathbb R\to\mathbb R$ be a convex function. Then it follows by Jensen's inequality that
$$\mathbb E[\psi(X)]~~\le~~ \mathbb E[\psi(Y)]$$
and if ...
3
votes
1
answer
186
views
Inequality of a concave function
Let $G:\mathbb R\to\mathbb R$ be a concave function, define $G_{\epsilon}: \mathbb R\to\mathbb R$ by
$$G_{\epsilon}(x)~~:=~~\max_{y\in [x-\epsilon, x+\epsilon]}G(y).$$
My question is the following: ...
4
votes
1
answer
4k
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Can you give me good examples of non-convex functions that are problematic for optimization?
I want to test my extended gradient descent algorithm, whose aim is to handle non-convex problems better. Can you give me some examples of non-convex functions that are hard to minimize via gradient ...
12
votes
3
answers
2k
views
Are small $\varepsilon$-balls convex in geodesic metric spaces?
Let $(M,d)$ be a complete, separable, compact metric space. Assume $M$ is geodesic, that is for any $x,y \in M$ there exists a distance realizing geodesic between $x$ and $y$ (not necessarily unique). ...