The convexity tag has no usage guidance.

**-1**

votes

**0**answers

51 views

### Is element-wise convex function convex? [closed]

Let $x$ be a $n$-dimensional vector, and $f(x)$ be a function of $x$.
If $f(x)$ is element-wise convex, is this function a convex function?

**0**

votes

**0**answers

52 views

### deriving concave upper bounds of a domain constrained nonconvex function over a simplex

Consider a nonconvex function $h(X)=f(X^\dagger AX)$, where $X\in C^{r \times n}$, $A$ is a positive semidefinite matrix, and $f$ satisfies the following two properties:
\begin{align}
&f(W): H_{+}^...

**0**

votes

**0**answers

26 views

### convex representation of a combinatorial constraint

I have an optimization problem with a weird constraint as follows. Is it possible to express it in some ways that have convex properties:
matrix $\mathbf{X}$ is either
$[1 \ 0 \ 0 \ 0 \ 0\\
\ 0 \ 0 ...

**2**

votes

**1**answer

110 views

### Carathéodory's theorem for $SO(3)$?

Let $Q \in \operatorname{Conv} SO(3)$.
Is there a way to retrieve an explicit representation of $Q$ as convex combination $Q=\sum_{k=1}^{r}{\lambda_{i}R_{i}},
R_{i} \in SO(3)$?
An approximation ...

**0**

votes

**0**answers

47 views

### Intersection Of Valentine Convex Sets

A set X is said to be m-convex , m integer >=2, if for each set of m points at least one of the associated line segments lies in X.
A 3-convex set is sometimes also known as Valentine convex after ...

**0**

votes

**2**answers

121 views

### Strictly convex norm on an infinite-dimensional Hilbert space

Question: Consider the Hilbert space $ H=\ell^2(\mathbb{Z})$. Let ${\rm L}(H)$ be the set of all linear
operators on $H$ onto itself. Find a norm $N$ and a domain $DN\subset {\rm L}(H)$ for $N$ ...

**1**

vote

**0**answers

58 views

### On convex hull of algebraic curve

For any given tuple of quadratic functions $(f_1(x,y,z),f_2(x,y,z),f_3(x,y,z))$, the following set forms a algebraic curves
$$
S_{f_1,f_2,f_3}=\{(f_1(x,y,z),f_2(x,y,z),f_3(x,y,z))|x^2+y^2+z^2=1,x,y,z\...

**0**

votes

**0**answers

40 views

### Limit of argmin of sum

Suppose that I know $f_n\rightarrow f$ and $g_n\rightarrow g$ are both continuous maps from a Complete Riemmanian Manifold $X$ to $\mathbb{R}$ which converge pointwise almost everywhere. Then is it ...

**0**

votes

**1**answer

78 views

### Showing convexity of a function in the unit ball

We have the unit sphere $S^2$ in $\mathbb{R}^3$ and two points, $X$ and $Y$ on the surface of the sphere. Then, a function is defined for any point $P$ inside of the unit ball as:
$$f(P) = R\,d(P, XY)...

**0**

votes

**0**answers

51 views

### Convexity condition of matrices

In studying the viscoelastic theory of elastodynamics, I encounter a problem on the convexity condition of matrix functions. It has been known that for the energy function $E=E(v,F) = \frac{1}{2} v^2 +...

**2**

votes

**0**answers

31 views

### Is there a “last mile” criterion for a generalization of planar convex hulls to symmetric weighted graphs?

This question is motivated by an almost successful attempt to define planar convex hulls of a finite set of isolated points only via subset sums of the distances between point-pairs; the advantage of ...

**1**

vote

**0**answers

57 views

### Hessian matrix positive definiteness (concavity test) [closed]

I have a rather simple scenario based optimization problem:
Maximize
$$
Q_1{_s}(A_1{_s}-Q_1{_s}-bQ_2{_s})+ Q_2{_s}(A_2{_s}-Q_2{_s}-bQ_1{_s})-[(Q_1{_s}-K_1)^+ + (Q_2{_s}-K_2)^+]c
$$
subject to $Q_1{...

**2**

votes

**1**answer

101 views

### Sufficient conditions for weak majorization

Suppose $x_1\ge x_2\ge \cdots \ge x_n\ge 0$ and $y_1\ge y_2\ge\cdots\ge y_n\ge0$ be reals such that for any positive integer $p$,
$$
\sum_{i=1}^n x_i^p \geq \sum_{i=1}^n y_i^p.
$$
Question: Is ...

**3**

votes

**1**answer

136 views

### Global minimum of nonlinear least square

We have a continuous and differentiable function $f(\cdot)$ that maps from $R^n$ to $R^n$. We are trying to solve a nonlinear least square problem:
Minimize $J(x)=\Vert f(x)-z\Vert^2$
subject to box ...

**9**

votes

**0**answers

154 views

### Determining convexity of a polygon from its Fourier coefficients

Consider an $n$-sided polygonal curve in the plane, represented by an ordered set of points $(x_0, x_1, \ldots, x_{n-1})$; line segments connect consecutive points and also $x_{n-1}$ to $x_0$. It is ...

**3**

votes

**2**answers

106 views

### Eigenspace of convex combination of two idempotent matrices

Let $H_1,H_2\in\mathbb{Q}^{n\times n}$ be idempotent and symmetric matrices. For any $0<\mu<\frac{1}{2}$, consider the matrix
$$H_\mu:=\mu H_1+(1-\mu)H_2.$$
I'm looking for a description of $\...

**0**

votes

**1**answer

41 views

### Representation of probability measure over product spaces

Trying to obtain some exchangeability-related results, I ended up with the following questions, which I couldn't answer (at least, not in the negative); this is also related to this MO thread (edit: ...

**8**

votes

**3**answers

271 views

### Shape whose translated and scaled copies are closed under intersection

The translated and scaled copies of an equilateral triangle with fixed orientation are closed under intersection - the intersection is again an equilateral triangle with the same orientation.
What ...

**1**

vote

**0**answers

42 views

### Calculating a Combinatorial Generalization of Planar Convex Hulls

In this question I have suggested a generalization of the notion of a set of points in the Euclidean plane being in convex configuration to the set of vertices of symmetric weighted graphs via $k$-...

**3**

votes

**1**answer

89 views

### Generating Uniquely k-Optimal Point Sets

This question is motivated by the observation that finding an optimal tour through a set of points in the Euclidean plane is especially simple, if the points are in convex configuration and, that the ...

**1**

vote

**0**answers

28 views

### Is support function of a convex curve in $\mathbb{R}^2$ absolutely continuous? [closed]

There is an example of a convex function of two variables that is convex but not absolutely continuous. The level set of this function is a convex curve.
To construct one example of such a function ...

**2**

votes

**1**answer

252 views

### Is this a log-concave function?

Let $(a_k)$ be a log-concave positive decreasing sequence. Is $\sum\limits_{k=1}^n a_k(1-e^x)^{k-1}$ log-concave in $x<0$, for each natural $n$?

**2**

votes

**0**answers

100 views

### Log-concave polynomial is a log-concave function?

A polynomial $\sum\limits_{k=0}^n a_kx^k$ is log-concave if $a_0,\ldots,a_n$ constitute a log-concave sequence. I wonder whether the log-concave polynomial is also a log-concave function with respect ...

**3**

votes

**1**answer

128 views

### The center of a minimal convex superbody

Is the following true?
CONJECTURE: $\,$ Let $\ B\ C\subseteq\mathbb R^n\ $ be convex bodies in $\mathbb R^n$ such that $\ C\ $ is centrally symmetric, $\ B\subseteq C,\ $ and $\ t\!\cdot\! B\ $ ...

**15**

votes

**1**answer

662 views

### On convergence of convex bodies

Let $K\subset \mathbb{R}^n$ be a compact convex set of full dimension. Assume that $0\in \partial K$.
Question 1. Is it true that there exists $\varepsilon_0>0$ such that for any $0<\...

**6**

votes

**1**answer

311 views

### Finding non convex functions satisfying a weak form of convexity, without the axiom of choice

If a real-valued function $f$ over reals satisfies $$ (1) \; \; \; f({x+y\over2})\le {f(x)+f(y)\over2}, $$and it is continuous, then it is not hard to see that $f$ is indeed convex. On the other hand,...

**4**

votes

**1**answer

202 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 ...

**6**

votes

**2**answers

278 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 $...

**5**

votes

**1**answer

93 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 \mathbb{R}...

**14**

votes

**0**answers

280 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 \...

**4**

votes

**1**answer

202 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 ...

**1**

vote

**0**answers

111 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 ...

**3**

votes

**1**answer

126 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 ...

**1**

vote

**1**answer

217 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

93 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 ...

**0**

votes

**0**answers

51 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
\begin{...

**2**

votes

**1**answer

118 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 = [0,1]\times[0,...

**5**

votes

**1**answer

85 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 ...

**5**

votes

**1**answer

139 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

**2**answers

99 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

377 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 f(\frac{x_1+x_3}2)+2\...

**4**

votes

**0**answers

77 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 ...

**5**

votes

**1**answer

124 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

333 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 ...

**4**

votes

**1**answer

181 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 ...

**7**

votes

**1**answer

255 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
$$\rho_2(X;Y):=\sup\frac{\mathbb{E}[f(X)g(Y)]}{||f||_2||...

**3**

votes

**1**answer

111 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

207 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 $f(x)=\min_{...

**3**

votes

**0**answers

117 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 |x_{ij}|...

**1**

vote

**1**answer

61 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 {f_2}\...