The convexity tag has no usage guidance.

**5**

votes

**1**answer

414 views

### Extreme points of a compact convex set are a $G_\delta$?

Dear All,
I'm reading a paper (Residuality of Dynamical Morphisms by Burton, Keane and Serafin) that makes a claim that I've been unable to verify or find a reference for. The claim is made that the ...

**3**

votes

**1**answer

119 views

### Mapping a subset of semi-definite matrices through arcsinus

Hi
I am meeting a problem concerning semi-definite positive matrices, and I have no clue concerning them, the classical approaches I know have not given any result, maybe people used to manipulating ...

**4**

votes

**1**answer

418 views

### Is it possible to extend this inequality about Euclidean distance &Frobenius norm to more general Bregman divergence such as relative entropy & von Neumann divergence?

Motivation- A Special Case
Supposing $A,B\in\mathbb{S}^{m\times m}$ are symmetric positive semi-definite (SPD) matrices and $\mathbf{x}\in\mathbb{R}^m$ is a unit vector where $\|\mathbf{x}\|=1$, we ...

**9**

votes

**3**answers

2k views

### Area Enclosed by the Convex Hull of a Set of Random Points

Consider $n$ points generated randomly and uniformly on a unit square. What is the expected value of the area (as a function of $n$) enclosed by the convex hull of the set of points?

**1**

vote

**1**answer

443 views

### conjugate function for matrix mixed norm

I am familiar with the conjugate function of the vector norm, which uses the concept of dual norm and is defined as follows:
...

**0**

votes

**1**answer

226 views

### Is there relationship between $f\left({\sum_i(\mathbf{v}_i^\top\mathbf{x})^2\lambda_i},\sum_j{(\mathbf{u}_j^\top\mathbf{x})^2\theta_j}\right)$ and $\sum_i\sum_j{(\mathbf{v}_i^\top\mathbf{u}_j)^2f(\lambda_i,\theta_j)}$ if $f$ is jointly convex?

Hello, everyone!
As we know that by Jensen's inequality, for jointly convex function $f$ and $\sum_ix_i^2=1$, we have
...

**2**

votes

**1**answer

262 views

### Generalized unique nearest point problem for a compact, convex set in a strictly convex Banach space.

If $X$ is a Real Banach space with strictly convex norm, it is known that for any non-empty compact, convex set $K$ and point $x_0\notin K$, there exists a unique point $z_0\in K$ minimizing the ...

**1**

vote

**0**answers

194 views

### Given a jointly convex function $f$, what is the bound of $f\left(\sum_ip_i^2x_i,\sum_jq_j^2y_j\right)$if $\mathbf{p},\mathbf{q}$ are constrained in a manifold?

Suppose there is a jointly convex function $f:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$, $\mathbf{x},\mathbf{y}\in\mathbb{R}^m$ and ...

**2**

votes

**1**answer

2k views

### Sufficient conditions for gradient descent convergence

I have an unconstrained optimisation problem with convex objective function $f(x)$. Suppose I have access only to some function of the gradient $\hat{\nabla}= g(\nabla f)$, and I take gradient steps ...

**3**

votes

**2**answers

1k views

### Computational complexity of unconstrained convex optimisation

What is known about the relationship between unconstrained convex optimisation and computational complexity? For example, for which optimisation problems and which gradient descent algorithms is one ...

**13**

votes

**2**answers

518 views

### Extreme points of unit ball in tensor product of spaces

Let $B_1, B_2$ be unit balls in finite-dimensional normed spaces $X_1, X_2$ respectively.
Let $e(B_1), e(B_2)$ be corresponding extreme points sets.
Consider the unit ball $B$ in tensor product ...

**2**

votes

**1**answer

675 views

### Local strong convexity of a strictly convex function

Is there a formal way to characterise strong convexity about the optimum value of a strictly convex function? I have an objective that looks something like this: $J(p,q) = ...

**0**

votes

**0**answers

207 views

### lipschitz property of the derivative of a convex function

Let $f\in C^1(\mathbb R^n\to \mathbb R)$ be a convex function. Suppose the equation
$$f(x+\Delta x)-f(x)-(f'(x),\Delta x) \leq A|\Delta x|^2$$
holds for some constant $A>0$, any $x\in \mathbb ...

**3**

votes

**3**answers

517 views

### Inequalities for uniformly convex normed spaces

When reading "Chebyshev centers and uniform convexity" by Dan Amir I encountered the following result which is apparently "known and easy to prove". I'm sure it is, but I can't find a proof and am ...

**8**

votes

**0**answers

310 views

### What is the “positive part” of the unit ball in $M_n(R)$ ?

In ${\bf M}_n(\mathbb R)$, let us consider the usual operator norm
$$\|A\|=\sup\frac{\|Ax\|}{\|x\|},$$
where $\|x\|$ is the Euclidian norm.
The closed unit ball $B$ is the set of contractions (in the ...

**3**

votes

**1**answer

500 views

### Tverberg partitions with less than (r-1)(d+1)+1 points

The Tverberg Theorem states the following: Let $x_1,x_2,\dots, x_m$ be points in $R^d$ with $m \ge (r-1)(d+1)+1$. Then there is a partition $S_1,S_2,\dots, S_r$ of $\{1,2,\dots,m\}$ such that $\cap ...

**2**

votes

**0**answers

260 views

### Erdős-Szekeres empty pseudoconvex $k$-gons

I am wondering if the
Erdős-Szekeres
empty convex $k$-gon question has a different answer if
convexity is replaced by a pseudoline-version of convexity.
The empty convex $k$-gon question
is a variant ...

**1**

vote

**0**answers

107 views

### Mappings preserving convex compactness

Let $H$ be a Hilbert space.
How can one describe continuous mappings $F:H \to H$
that satisfy the following condition:
There exist two elements $c$, $F(c) \neq c$
and a convex compact $M$ containing ...

**6**

votes

**3**answers

2k views

### A criterion for the sum of two closed sets to be closed ?

Let $V$ and $I$ be two closed subsets of a Banach space $A$.
The set $V$ is a convex cone, and $I$ is a linear subspace of $A$. I also know that $V\cap I=\{0\}$.
I would like to know whether $I+V$ ...

**21**

votes

**1**answer

499 views

### The maximal “nearly convex” function

The following problem is only tangently related to my present work, and I do
not have any applications. However, I am curious to know the solution -- or
even to see a lack thereof, indicating that the ...

**4**

votes

**1**answer

360 views

### Generalized widths and reverse Urysohn inequalities

This question is inspired by the discussion in MO questions "Local minimum from directional derivatives in the space of convex bodies" and "Bodies of constant width?" about generalized notions of ...

**5**

votes

**0**answers

284 views

### Local minimum from directional derivatives in the space of convex bodies

I have a function $f(K)$ defined on the space of three-dimensional convex bodies for which I want to show that the unit ball $B$ is a local minimum. I have been able to show if $K$ is not homothetic ...

**3**

votes

**0**answers

164 views

### Characterizing curves that bound strictly convex regions

Consider a closed curve on the plane so that if we perform any translation and dilation on it, the resulting curve intersects the original curve at most twice. Does this property characterize the ...

**6**

votes

**2**answers

322 views

### Volume inequality between projections of a convex symmetric set in $\mathbb R^3$

Let $K$ be a centrally symmetric convex body in $\mathbb R^3$ with volume ${\rm vol}(K)=1$. For any subset $F \subset \lbrace1,2,3\rbrace$, let $K_F$ be the projection of $K$ in $\mathbb R^F$.
...

**8**

votes

**1**answer

1k views

### Extreme points of a set of probability measures

Consider the set of Borel-measurable probability measures over the interval $[0,1]$ with a given mean, say 1/2. To be precise, I'm talking about the following set $$M=\left(\mu\in \Delta([0,1]):\int ...

**1**

vote

**1**answer

601 views

### Shift invariant measures that are(n't) convex combinations of ergodic measures

Let $X=2^\omega = \lbrace 0,1 \rbrace^{\mathbb{N}}$ and $T\colon X \to X$ the (left) shift map. The space $\mathcal{M}$ of $T$-invariant Borel probability measures is convex with $\mathcal{M}^e$, the ...

**2**

votes

**1**answer

464 views

### minimizing functions over simple matrix inequalities

I'm wondering if anything is known about minimizing convex, not necessarily linear functions subject to "simple" matrix equalities. To be precise, consider the following example:
$min \Sigma x_i ln ...

**8**

votes

**2**answers

215 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 ...

**2**

votes

**1**answer

141 views

### Spline fit with bounded derivations

How can I do a Spline Fit with bounds on some derivations?
Problem
Given:
Set of data points $t_k, x_k$
Set of nodes $n_i$
order $D$ of the spline (in my case $D=5$)
lower and upper bounds ...

**11**

votes

**1**answer

408 views

### convexity of images of space-filling curves

Suppose $f:[0,1]\to[0,1]^2$ is continuous and for each $t\in[0,1]$, the area of $\lbrace f(s) : 0\le s\le t \rbrace$ is $t$. For what sets of values of $t\in[0,1]$ can $\lbrace f(s) : 0\le s\le t ...

**8**

votes

**2**answers

971 views

### Does a notion of convex graph make sense?

Let $X=(V,E)$ be a finite connected graph. I would be interested in some notion of convexity.
General question: Is there a notion of convexity for finite connected graphs? How does it look like?
...

**3**

votes

**1**answer

576 views

### Java library for SDP [closed]

People who frequently code semi definite programs, is there any java library for solving sdps? I have tried my luck but all I can find is C/C++ libraries or matlab toolboxes. I can write wrappers to ...

**4**

votes

**2**answers

651 views

### gradient of convex functions

Hello. Can somebody help me with the following question that I have thought over for quite some time, to no avail?
Let $f$ be a smooth function (class $\mathrm{C}^{\infty}$), $f:\mathbb{R}^n ...

**13**

votes

**2**answers

3k views

### Weighted area of a Voronoi cell

Let $X = \{ x_1,\dots,x_n\} $ denote a set of $n$ points in the unit square $S = [0,1]\times[0,1]$, and let $w = \{w_1,\dots,w_n\}$ denote a set of weights corresponding to the $n$ points in $X$. ...

**3**

votes

**1**answer

239 views

### Signed measure that is positive over convex sets

I have a signed measure $\mu$ on a convex subset $C\subset \mathbb{R}^n$, and I want to prove that $\mu$ is a probability measure, most importantly that it is positive everywhere.
I do know that ...

**2**

votes

**0**answers

148 views

### Minimally 6-connected 3D discrete lines that are convex lattice sets

There are several definitions of 3D discrete lines, e.g. http://diwww.epfl.ch/w3lsp/publications/discretegeo/nratddl.html , http://dx.doi.org/10.1007/978-3-642-19867-0_4 . However, I know of none that ...

**3**

votes

**2**answers

212 views

### Does the automorphism group of a cone determine the cone?

A cone is a $R_+$-module. That is, a cone is an abelian monoid that is closed under nonnegative real scalar multiplication. An automorphism of a cone is a bijective $R_+$-linear map. That is a map ...

**2**

votes

**1**answer

307 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 $\langle A,B\rangle = tr (AB)=tr(BA)$.
...

**11**

votes

**1**answer

429 views

### Status of the compact AR problem?

The so-called "compact AR Problem" reads:
Is every compact convex set in a metrizable topological vector space an absolute retract?
It is open according to the chapter by T. Banakh, R. Cauty ...

**2**

votes

**2**answers

646 views

### Finding smallest ellipsoid that circumscribes over intersection of two ellipsoids that do not have common center

Does this already exist in literature? The closest Ive been able to find is circumscribe intersection of two ellipsoids with a common center by W. Kahan ...

**2**

votes

**1**answer

1k views

### How do I optimize over (or take derivative wrt) a square diagonal matrix?

Hello. I'd like to solve the following optimization problem.
$P_i$ is a 6x6 matrix
$X$, $Y$ is a 6xk matrix
$w_i$ is a kx1 vector
$diag(w_i)$ is a square diagonal matrix with diagonal entries equal ...

**0**

votes

**1**answer

163 views

### Can extremal matrices of subcones of psd matrices have low rank?

Let $S$ be the cone of positive semidefinite symmetric real matrices of size $n\times n$. The cone $S$ spans a $d:=n(n+1)/2$ dimensional vector space.
Let $C\subset S$ be a subcone formed by ...

**3**

votes

**2**answers

546 views

### Is a solution of a linear system of semidefinite matrices a convex combination of rank 1 solutions?

The cone of symmetric positive semidefinite $n\times n$ matrices is the convex hull of rank $1$ matrices. That is, every symmetric positive semidefinite matrix is a convex combination of rank 1 ...

**8**

votes

**0**answers

247 views

### Are plactic classes convex under the right weak Bruhat order?

For those who are unfamiliar with the terminology, I'll explain a little.
The symmetric group $S_n$, as a type A Coxeter group, has generators $\{s_1,\ldots,s_{n-1}\}$ with relations (1) $s_i^2$ for ...

**3**

votes

**2**answers

478 views

### Surface fitting with convexity requirement

Hi all,
Consider a cloud of points in 3D space (x,y,z). The data is well-behaved, once plotted the surface looks like some sort of spheroid. I assumed a form for the fitting function f(x,y,z) = c1 ...

**1**

vote

**0**answers

332 views

### Computing the intersection of dual affine subspaces

Suppose we have a convex function , $\phi(x): R^d \to R$. It is well known that the Legendre transform of $\phi$ is also a convex function, and can (loosely) be thought of as the dual or derivative ...

**1**

vote

**0**answers

184 views

### Joint Convexity of Spectral functions of several matrices

$\{A_1 \ldots A_K \}$ is a set of matrices in $\mathbb{R}^{m \times n}$. Let $f (A_1,\ldots,A_K)$ be a function of the singular values of all matrices. For e.g., $f$ is just summation of singular ...

**0**

votes

**0**answers

345 views

### Global index of convexity/concavity of a function

We are looking for a global index of the convexity/concavity of a function.
For concreteness, how can I formalize the intuitive notion that a function $f$ is more convex than $g$ where ...

**15**

votes

**2**answers

683 views

### Do the elementary properties of mixed volume characterize it uniquely?

Background
Take 2 convex sets in $\mathbb{R}^2$, or 3 convex sets in $\mathbb{R}^3$, or generally, $n$ convex sets in $\mathbb{R}^n$. "Mixed volume" assigns to such a family $A_1, \ldots, A_n$ a ...

**4**

votes

**1**answer

311 views

### Wasserstein geometry of measures on manifolds related to the generalized Legendre transform and $d^2/2$-convexity

Let $(M,g)$ be a fixed closed Riemannian manifold, normalized to have volume 1. We'll write $d_M(x,y)$ for the (geodesic) distance between two points $x,y\in M$. I'm interested in the following class ...