The convexity tag has no usage guidance.

**4**

votes

**1**answer

121 views

### Questions about the regularity of the “norm” associated to a convex set

Suppose $K\subset \mathbb{R} ^n$ is a closed convex set whose interior contains the origin. We can assign a gauge function to $K$ as $g_{K}(x):=\inf\{\lambda>0 \mid x\in\lambda K\}$. $g_K$ has all ...

**7**

votes

**1**answer

202 views

### A Generalized Version of Maximal Correlation and Hypercontractivity of Conditional Expectation Operator

Given a pair of random variables $(X,Y)$ over a product space $\mathcal{X}\times \mathcal{Y}$, the maximal correlation coefficient is defined as
...

**5**

votes

**2**answers

185 views

### Is every open convex subset of a Riemannian manifold necessarily contractible?

Question: Is every open convex subset $C$ of a Riemannian manifold $M$, necessarily contractible?
Here by a "convex subset" I mean a set $C$ having the property that between each pair of points in ...

**3**

votes

**2**answers

396 views

### Non-asymptotic large deviations for a convex set

Let $X_1,\dots,X_n$ be $n$ i.i.d random variables taking values in a Polish vector space $\mathcal{X}$ and with (Borel) probability distribution $\mu$.
For any convex, compact $\Gamma \subset ...

**13**

votes

**0**answers

214 views

### A functional inequality about log-concave functions

Let $f,g$ be smooth even log-concave functions on $\mathbb{R}^{n}$, i.e.,$f=e^{-F(x)}, g=e^{-G(x)}$ for some even convex functions $F(x),G(x)$. Is it true that:
$$
\int_{\mathbb{R}^{n}} \langle ...

**5**

votes

**1**answer

77 views

### Two (new?) variants of convex functions

I find that the following two types of functions are useful to my research.
(i) We know that a function $f: \mathbb{R}_+^m\rightarrow \mathbb{R}$ is called convex if for all ${\bf x,y}\in ...

**0**

votes

**1**answer

76 views

### sum of fractional functions optimization problem

Consider the following sum of fractional functions optimization problem
$$
\begin{array}{l}
\mathop {\min }\limits_{\bf{x}} \,\,\,\sum\limits_{i = 1}^p {\frac{1}{{{\bf{a}}_i^T{\bf{x}} + {b_i}}}} \\
...

**50**

votes

**1**answer

641 views

### Continuous maps which send intervals of $\mathbb{R}$ to convex subsets of $\mathbb{R}^2$

Let $f : \mathbb{R} \longrightarrow \mathbb{R}^2$ be a continuous map which sends any interval $I \subseteq \mathbb{R}$ to a convex subset $f(I)$ of $\mathbb{R}^2$. Is it true that there must be a ...

**4**

votes

**1**answer

118 views

### Maximal cones and lexicographic orderings

Let $V$ be a real vector space. It is well known that, given a totally ordered basis of $V$ (say $(b_i)_{i\in I}$ where $I,<$ is totally ordered), $V$ is totally ordered by the lexicographic ...

**3**

votes

**1**answer

111 views

### Generalization of standard convex problem

In standard convex programming, the objective function and each of the constraint inequalities are convex. in such case, if the KKT condition hold for a point, and Slater condition is also hold for ...

**4**

votes

**1**answer

177 views

### Is it possible to solve the argument maximization problem $\arg\max_x \langle x,l \rangle −f_1(x)−f_2(x)$ via convex duality?

I am attempting to solve the argument maximization problem
$$\arg\sup_x \{ \langle x,l \rangle − f_1(x)−f_2(x) \} \ \ \ \ \ \ \ \ \ \ (1)$$
where the functions $f_1$ and $f_2$ are concave but ...

**0**

votes

**0**answers

21 views

### Duality for Generalization of standard Convex

in accordance to the previous question about KKT condition for generalization to standard convex, here I look for the dual problem to the generalized convex problem. the clear questions are :
is it ...

**11**

votes

**2**answers

772 views

### Approximating a convex function by a piecewise linear function

Suppose I have a Lipschitz-continuous convex function $f:\mathbb{R}^n\rightarrow \mathbb{R}$. I wish to approximate it on the unit ball by a piecewise-linear function $g:\mathbb{R}^n\rightarrow ...

**4**

votes

**0**answers

105 views

### Symmetry of Sundry Planar Convex Sets of Constant Width & Minimal Area

In a much broader paper in “Optimization Methods & Software 27,6 (2012) pp1073-1099” Bayen & Henrion consider planar compact, convex sets with support functions which are finite Fourier ...

**0**

votes

**0**answers

73 views

### Proving that a complicated function is concave or strongly uni-modal

I am trying to prove the concave property for a complicated function during my research project (imperfect maintenance modelling for starter) which has the following form:
$$
\eta(t)= \alpha \beta ...

**1**

vote

**0**answers

38 views

### 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 ...

**0**

votes

**0**answers

94 views

### does every compact convex set in c0 have but countably many extreme points

This seems plausible, given the properties of the unit ball of $c_0$.
I have a compact set in a complex Banach space $X$ whose closed convex hull has uncountably many extreme points. It would be ...

**6**

votes

**0**answers

72 views

### Convexity of Isoperimetric Domains

I am interested in what is known about the convexity of isoperimetric domains in compact Cartan-Hadamard manifolds (Riemannian manifolds that are complete and simply-connected and have non-positive ...

**2**

votes

**1**answer

304 views

### Linear and Isometric Automorphism Groups of the PSD Cone

Let $S_+$ be the cone of psd matrices ($n\times n$ real symmetric positive semidefinite matrices). This cone is a metric space induced from the inner product $\langle A,B\rangle = tr (AB)=tr(BA)$.
...

**0**

votes

**0**answers

38 views

### Log-concave distributions: Weighted sum of pdfs

Assuming $f_n(\cdot)$ is a log concave function (e.g., pdf of Gaussian distribution) and $0\le q_n\le 1$ for all $n\le N$, I am trying to find conditions under which the following holds
...

**1**

vote

**0**answers

59 views

### Choosing the weights of a Voronoi diagram — is this function always the gradient of another function?

This question is related to the earlier question Weighted area of a Voronoi cell . As in that question, let $X = \{ x_1,\dots,x_n\} $ denote a set of $n$ points in the unit square $S = ...

**5**

votes

**1**answer

68 views

### when does elementwise-log preserve positive-semidefiniteness?

Let $Z$ be a positive semidefinite matrix with nonnegative entries, and define $X=\log(1+Z)$, where the $\log$ is taken entrywise, i.e., $X_{ij}=\log(1+Z_{ij})$. Are there some simple sufficient ...

**4**

votes

**1**answer

654 views

### Why are all these families of polynomials finally log-concave?

This started when I was examining certain families of unimodal polynomials, i.e. $\sum_{k=0}^n a_kx^k$ where $a_0\le a_1\le\cdots \le a_k\ge\cdots \ge a_n$.
(Notation: in the following, the $a_k$ ...

**2**

votes

**7**answers

4k views

### Existence of an extreme point of a compact convex set

The Krein-Milman theorem shows that a compact convex set in a Hausdorff locally convex topological vector space is the convex hull of its extreme points.
It seems this implies that a compact convex ...

**5**

votes

**1**answer

137 views

### (A kind of) Irreducibiliy of regular open convex sets in the Cartesian space

I am looking for a proof of the fact which is formulated at the bottom of this post. The property of regular convex sets which the fact expresses seems to be true to me, yet I have not been able to ...

**2**

votes

**1**answer

72 views

### Fixed point iteration on symmetric biconvex function

Suppose $X\subseteq\mathbb{R}^n$ is a convex set and that a function $g(x,y):X\times X\rightarrow\mathbb{R}_+$ is smooth, "strictly biconvex" (strictly convex in $x$ and $y$ independently but not ...

**8**

votes

**1**answer

330 views

### Generalization of Popoviciu's inequality

Popoviciu's inequality states that for convex $f$ and numbers $x_1,x_2,x_3$, we have
$f(x_1)+f(x_2) + f(x_3) + 3\cdot f(\frac{x_1+x_2+x_3}3) \geq 2\cdot f(\frac{x_1+x_2}2)+2\cdot ...

**4**

votes

**0**answers

65 views

### Linear projections of convex sets with unique preimages of boundary points

Fix a compact convex subset $C \subset \mathbb{R}^n$ with nonempty interior. For any subspace $S \subset \mathbb{R}^n$, let $P_S$ denote the orthogonal linear projection onto $S$. I'd like to claim ...

**2**

votes

**0**answers

144 views

### 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 ...

**5**

votes

**1**answer

119 views

### TVS with null topological dual space

In that post, I give an example of a TVS for which the topological dual is equal to $0$. But in the example, there is no open convex subset different from the empty set or the space itself.
Do you ...

**5**

votes

**2**answers

304 views

### Minimum of squared sum minus sum of squares

I know that
$$
\min_{\|x\|_2=1=\|y\|_2} \left(\sum_{k=1}^nx_ky_k\right)^2-\sum_{k=1}^nx_k^2y_k^2 \geq -1/2
$$
with equality whenever $|x_k|=\frac{1}{\sqrt{2}}=|y_k|$ for two coordinates.
I'm ...

**3**

votes

**1**answer

153 views

### Subconvexity bound for Hecke $L$-functions in the $s$-aspect

Let $L(s,\chi)$ be the $L$-function of a non-trivial Hecke character of a general number field $K$, so that $L(s,\chi)$ which has no pole or zero at $s=1$.
I am looking for a reference for upper ...

**3**

votes

**1**answer

96 views

### Gradient estimate of convex functions

Consider a special type of convex function $g(\cdot):\mathbb{R}^d \to \mathbb{R}_+\cup\{+\infty\}$ such that $g(x)=+\infty$ as $|x|\to \infty$. Then $g$ is differentiable almost everywhere within its ...

**8**

votes

**0**answers

199 views

### Less complicated proof of this “obvious” fact about convexity

Let $C\subset \mathbb{R}^n$ be a compact, convex set. In any convex analysis course, it would be a standard homework exercise to prove that the functions $f(x)=\max_{y\in C} \|x-y\|$ and ...

**3**

votes

**0**answers

81 views

### On the modulus of convexity of mixed-norm $\ell_{p_1,p_2}$ spaces

Let $\ell_{p_1,p_2}=(\mathbb{R}^{m\times n},\|\cdot\|_{p_1,p_2})$ be the space of $m\times n$ matrices endowed with the mixed-norm
$$ \|X\|_{p_1,p_2} = \left( \sum_{j=1}^n \left( \sum_{i=1}^m ...

**1**

vote

**1**answer

55 views

### monotonicity alike functions

assuming we have two smooth function ${f_1},{f_2}:{R^N} \to R$,
under what condition, we have
${f_1}\left( {{{\bf{x}}_1}} \right) \ge {f_1}\left( {{{\bf{x}}_2}} \right) \leftrightarrow ...

**0**

votes

**0**answers

62 views

### Is the function below convex?

I have the following function $f(X)=(\sum(gmm^2(AX)-2gmm(AX)gmm(B)))||CX-D||^2$
where gmm is Gussian mixtures defined as $gmm(x)=\sum_{i=1}^{K}\omega_{i}\phi(x|\mu_{i},\Sigma_{i})$, $\omega$ is the ...

**1**

vote

**0**answers

95 views

### 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 ...

**2**

votes

**0**answers

97 views

### How to check if a manifold can be foliated by strictly convex hypersurfaces?

Let $M$ be a compact Riemannian manifold with boundary.
How can one recognize whether the manifold can be foliated by strictly convex hypersurfaces?
An exact definition is given below.
If the ...

**6**

votes

**3**answers

647 views

### Some questions about Invexity

Recently, I am looking into a non-convex optimization problem whose points satisfying KKT conditions can be obtained. Then the problem becomes how to decide whether the KKT conditions are sufficient ...

**2**

votes

**0**answers

122 views

### 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
...

**10**

votes

**2**answers

389 views

### Inequalities for averaging over partially ordered sets

Let's start from a classical inequality:
If $0\le a_1\le\cdots\le a_k$ and $0\le b_1\le\cdots\le b_k$ then
$(a_1+\cdots+a_k)(b_1+\cdots+b_k)\le k(a_1b_1+\cdots+a_k b_k)$.
It can be written also in ...

**1**

vote

**1**answer

92 views

### What's the most efficient way to solve this euclidean projection on non-negative affine space constraint?

I've come across this convex optimization problem in my research where I need to project a matrix $X_0$ onto a non-negative affine space constraint and box constraints. Concretely,
$X \in ...

**1**

vote

**0**answers

49 views

### Conditions on probability measure that generates non-void random polytope

Let $C$ be a non-void compact convex set in $\mathbb{R}^d$, and $\nu$ a probability measure on $C$. Then under what conditions on $C$ and $\nu$, the following statement is true: If ...

**4**

votes

**1**answer

216 views

### Orlicz Norm and A result on expectation

I am reading paper which is mainly about Dobrushin's contraction coefficient and its generalization. In page 27, the following is defined:
Consider arbitrary, non-negative, convex function ...

**2**

votes

**1**answer

101 views

### Fastest 'Oracle' Algorithm for satisfying a single linear constraint on a convex set?

In this paper by Arora, Hazan, and Kale (http://www.cs.princeton.edu/~arora/pubs/MWsurvey.pdf) a method is given for using the Multiplicative Weights Update algorithm to quickly solve Linear Programs ...

**22**

votes

**13**answers

4k views

### Generalizations of the Birkhoff-von Neumann Theorem

The famous Birkhoff-von Neumann theorem asserts that every doubly stochastic matrix can be written as a convex combination of permutation matrices.
The question is to point out different ...

**5**

votes

**1**answer

347 views

### Examples of Polyhedra with Large Shadows

Let $P \subseteq \mathbb{R}^n$ be a polyhedron described by $\mathcal{O}(n^{c_1})$ inequalities, where $c_1$ is a constant. Moreover, let $M\colon P \to \mathbb{R}^2$ be a linear mapping. I'm looking ...

**0**

votes

**0**answers

28 views

### Looking for a homogeneous function with some properties

I'm looking for a 1-homogeneous function $\pi \colon \mathbb{R}^n_{\geq 0} \to \mathbb{R}$ satisfying the following properties:
1) $\pi$ is not concave. This is equivalent to the fact that there ...

**3**

votes

**0**answers

180 views

### Convex Conjugate of Relative Entropy

The convex conjugate of a function, say, $f:X\mapsto \mathbb{R}$ is a function $f^*:X^*\mapsto \mathbb{R}$ defined as
$$f^*(x^*):=\sup_{x\in X} ~\langle x, x^*\rangle-f(x),$$ where $X^*$ is the ...