Questions tagged [convexity]

For questions involving the concept of convexity

Filter by
Sorted by
Tagged with
2 votes
1 answer
69 views

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 \...
Hans's user avatar
  • 2,863
8 votes
1 answer
272 views

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 $\...
Taras Banakh's user avatar
  • 40.7k
3 votes
1 answer
87 views

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 ...
Tychonoff's user avatar
1 vote
1 answer
294 views

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 ...
groupoid's user avatar
  • 580
1 vote
1 answer
103 views

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 $$\...
xFioraMstr18's user avatar
6 votes
3 answers
810 views

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, ...
Iosif Pinelis's user avatar
4 votes
0 answers
68 views

"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
  • 51.5k
8 votes
2 answers
675 views

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 ...
Janko Bracic's user avatar
2 votes
0 answers
316 views

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) \...
Manfred Weis's user avatar
  • 12.6k
10 votes
0 answers
252 views

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 ...
Mohammad Ghomi's user avatar
37 votes
0 answers
1k views

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 ...
Mohammad Ghomi's user avatar
2 votes
0 answers
81 views

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 ...
Martha Łącka's user avatar
7 votes
3 answers
418 views

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 ...
Ivan's user avatar
  • 689
24 votes
3 answers
1k views

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 ...
zref's user avatar
  • 343
7 votes
1 answer
135 views

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 ...
Jairo Bochi's user avatar
  • 2,411
1 vote
1 answer
71 views

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 ...
user2379888's user avatar
16 votes
4 answers
1k views

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 ...
MCH's user avatar
  • 1,304
1 vote
0 answers
88 views

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?
Turbo's user avatar
  • 13.6k
1 vote
0 answers
146 views

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 ...
JYY's user avatar
  • 133
1 vote
1 answer
582 views

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:...
Boby's user avatar
  • 631
7 votes
0 answers
869 views

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 ...
asv's user avatar
  • 21.1k
3 votes
1 answer
604 views

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 ...
Boby's user avatar
  • 631
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
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 ...
ernilesh80's user avatar
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 ...
KTU's user avatar
  • 161
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<...
Jacob's user avatar
  • 63
5 votes
1 answer
519 views

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 ...
asv's user avatar
  • 21.1k
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 ...
user_lambda's user avatar
1 vote
1 answer
2k views

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 $f$ ...
AvidLearner's user avatar
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)=\...
user_lambda's user avatar
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 ...
user_lambda's user avatar
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 ...
Jairo Bochi's user avatar
  • 2,411
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 $\...
M. Winter's user avatar
  • 12.5k
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 ...
Turbo's user avatar
  • 13.6k
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 ...
erz's user avatar
  • 5,295
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 ...
Grove's user avatar
  • 91
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;...
erz's user avatar
  • 5,295
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\...
Sven Pistre's user avatar
18 votes
1 answer
2k views

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,...
Tomás's user avatar
  • 409
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(...
Taylorien's user avatar
  • 131
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 ...
Bazin's user avatar
  • 15k
38 votes
2 answers
2k views

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 ...
erz's user avatar
  • 5,295
4 votes
1 answer
254 views

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 ...
user00169's user avatar
1 vote
1 answer
70 views

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 ...
Vanessa's user avatar
  • 1,368
44 votes
7 answers
3k views

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)=\...
T. Amdeberhan's user avatar
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 $f$...
Khadija Mbarki's user avatar
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 ...
CodeGolf's user avatar
  • 1,837
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: ...
CodeGolf's user avatar
  • 1,837
4 votes
1 answer
4k views

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 ...
Brans's user avatar
  • 141
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). ...
quarague's user avatar
  • 622

1
4 5
6
7 8
13