Questions tagged [convexity]

For questions involving the concept of convexity

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0answers
113 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
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1answer
555 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
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1answer
55 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 ...
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0answers
48 views

Conditional moments estimate

Let $V\in C^\infty(\mathbb R^{d_1}\times \mathbb R^{d_2};\mathbb R)$, such that $Hess V(x,y)\geq \alpha\,I$ for some $\alpha>0$ (namely $V$ is uniformly convex). Thus $V$ as a unique minimum point $...
-4
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1answer
84 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 ...
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0answers
114 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
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0answers
89 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 $X'$ ...
3
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0answers
27 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 \...
8
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0answers
126 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 ...
9
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1answer
135 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 $K$-...
49
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2answers
4k 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
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0answers
48 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 ...
3
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0answers
145 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 ...
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0answers
46 views

convexity of a function inside expected value

Assume that $g(x)=\int_{y\in Y}h(y)f(x,y)dy$. I know the properties of function $g(\cdot)$, e.g. first- and second-order derivatives signs. Is there a way to drive the same properties i.e. convexity ...
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0answers
36 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 ...
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0answers
34 views

Convex separation in CAT(0) spaces

Are there any convex separation theorems for CAT(0) spaces? Particularly, can one separate two disjoint convex sets by a horosphere? In Riemannian manifolds, I have seen some results on convex ...
3
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0answers
46 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. ...
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0answers
16 views

Convexity and supermodular functions

Suppose that the function F(x,y) is supermodular in (x,y). We know that this implies that x* =argmax F(x,y) is increasing in y. Under which conditions (if any) we have that x* is a convex function of ...
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0answers
49 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 $$ \...
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1answer
48 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 ...
14
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3answers
731 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-...
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0answers
46 views

Second-order def of strong convexity wrt general norms

A function $f: R^n \rightarrow R$ is said to be C-strongly convex with respect to a norm $\|\cdot\|$ if for all $x,y$ and $\lambda \in [0,1]$ $$f(\lambda x + (1-\lambda)) \le \lambda f(x) + (1-\lambda)...
1
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1answer
72 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
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1answer
101 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
2answers
427 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?
2
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1answer
130 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 the Slater's inequality (compagnion of Jensen's inequality) I find this statement : Let $f(x)$ be a continuous,twice differentiable function ,convex or concave and non constant on $(0,\...
1
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1answer
37 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
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0answers
304 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 ...
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0answers
60 views

Characterization of gradient of convex conjugate

Let $X$ be a separable Hilbert space. Is there a known characterization of all functions $F:X\rightarrow X$ for which there exists a proper (not identically $\infty$), lower semi-continuous convex ...
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0answers
15 views

Can planar convexity be decided with matching

This question actually aims at generalizing the notion of a finite set of points in the euclidean plane being in convex configuration to complete , undirected simple graphs with arbitrary edge-weights....
9
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1answer
232 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
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0answers
64 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 ...
3
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0answers
34 views

Self-duality of cones associated with elementary symmetric polynomials

Let $n\ge3$ be an integer, and denote $\sigma_1,\ldots,\sigma_n$ the elementary symmetric polynomials in $n$ indeterminates: $$\sigma_1(X)=X_1+\cdots+X_n,\quad\ldots\quad,\sigma_n(X)=X_1\cdots X_n.$$ ...
0
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0answers
73 views

About the definition of lineal convexity

I have been trying to understand the definition of lineal convexity. I am reading the article Duality of functions defined in lineally convex sets by Christer O. Kiselman. For a set $A\subset \mathbb{...
0
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1answer
102 views

Expectation of random matrix

Assume $Q$ is a positive definite random matrix such that $0 < \lambda_{\min}(Q)....\leq \lambda_{\max}(Q) \leq 1$ holds. I want to show that \begin{align} E\left[\frac{\lambda_{\min}(Q)}{\lambda_{\...
1
vote
1answer
143 views

Projection of convex set onto a convex set [closed]

Can projection of convex sets onto convex sets be non-convex yet connected? If so is there any necessary and sufficient conditions? Can projection of $n$ dimensional convex sets in $\mathbb R^{n'}$ ...
2
votes
2answers
190 views

A question about asymptotic affinity and strict convexity with unbounded means

Let $F:[0,\infty) \to [0,\infty)$ be a $C^1$ strictly convex function. Let $\lambda_n \in [0,1],a_n\le c<b_n \in [0,\infty)$ satisfy $$ \lambda_n a_n +(1-\lambda_n)b_n=c_n \tag{1}$$ and assume that ...
4
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1answer
154 views

Does strict convexity plus asymptotic affinity imply bounded mean?

I am not sure if this is exactly research-level, but I am struggling to find a proof for the following claim: Let $F:[0,\infty) \to [0,\infty)$ be a $C^2$ strictly convex function. Let $\lambda_n \in [...
5
votes
1answer
269 views

Closed convex hull in infinite dimensions vs. continuous convex combinations

tl;dr: When is the closed convex hull of a set $K$ equal to the set of "continuous" convex combinations of $K$? I am essentially asking for the most general, infinite-dimensional analogue of ...
1
vote
1answer
180 views

Is this $(\Bbb R^{n \times n})^n \to \Bbb R$ function convex?

Let $W := (W_1, W_2,\dots, W_n)$, where $W_i \in \Bbb R^{n \times n}$. Let $x$ be a constant vector. Is the following function convex? $$f(W) := x^TW_1^TW_2^T \cdots W_n^TW_n \cdots W_2W_1x $$
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0answers
119 views

Log-concavity inequality

Let $x,y,$ and $t$ be fixed real numbers, $1<x<y$, $0<t<1$. Does the following inequality hold for some $c$ $$\frac{\log{(tx+(1-t)y)}}{\log^t{x}\log^{(1-t)}{y}}>\frac{\log{(sw+(1-s)z)}}{...
7
votes
2answers
395 views

Is a function of several variables convex near a local minimum when the derivatives are non-degenerate?

This is a cross-post. Let $U \subseteq \mathbb R^n$ be an open subset, and let $f:U \to \mathbb R$ be smooth. Suppose that $x \in U$ is a strict local minimum point of $f$. Let $df^k(x):(\mathbb R^n)^...
2
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0answers
52 views

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 $M$ ...
4
votes
5answers
590 views

Elementary inequality generalizing convexity of a function on a segment

I am looking for a proof of the following statement which is known to be true as far as I heard. Let $g\colon [a,b]\to \mathbb{R}$ be a smooth function. Assume that $$b-a< \pi.$$ Assume also $$g(a)\...
1
vote
1answer
138 views

Baffling proof using function convexity

Let the function $f: \mathbb{R} \rightarrow \mathbb{R}$ be convex, differentiable with derivative $f_x$ and Lipschitz continuous with constant $L$. Then, for $a,b,c,d \in \mathbb{R}$ such that $a \ge ...
0
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1answer
110 views

Prove the optimal solution to maximizing nuclear norm with constraints is attained at corner points of feasible region

The nuclear norm (trace norm) of a matrix $X \in \Bbb R^{m \times n}$ is defined as $$\|X\|_* := \sum_{i=1}^{\min(m,n)} \sigma_i(X)$$ where $\sigma_i(X)$ are the singular values of $X$. The ...
1
vote
1answer
181 views

Convexity at a point and Jensen inequality

I am looking for a reference for the following claim: Let $\phi:\mathbb (a,b) \to \mathbb R$ be a continuous function, and let $c \in (a,b)$ be fixed. Suppose that "$\phi$ is convex at $c$". ...
2
votes
1answer
105 views

Lower bound on $L^2$ norm of a strongly convex function

Let $f\colon[0, 1] \to \mathbb R$ be an $m$-strongly convex function and $\mu$ be a probability measure on $[0,1].$ For any $t<1$, the goal is to find a lower bound on $\int_{0}^t f^2(x) d\mu(x)$ ...
0
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0answers
34 views

Quasi-concavity of minimum of function

Consider a differentiable function $F(x,y,z)$ defined on $[0,1]\times[0,1]\times[0,1]$, which is increasing and quasi-concave in (x,y). That is, the partial derivatives of $F$ with respect to $x$, $y$ ...
0
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0answers
54 views

The scalar convergence in $\mathcal{C}(X)$ is topologizable?

Let $(X,\|.\|)$ be a separable Banach space and $\mathcal{C}(X)$ be the collection of all nonempty, closed and convex subsets of $X$. For any $C$ in $\mathcal{C}(X)$ we set $$ s(x^*, C) := \sup_{x\in ...

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