The convexity tag has no wiki summary.

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225 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 ...

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**0**answers

141 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**

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**2**answers

207 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 ...

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votes

**1**answer

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

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**1**answer

423 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 ...

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votes

**2**answers

519 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 ...

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votes

**1**answer

869 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 ...

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**1**answer

157 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 ...

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**2**answers

452 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 ...

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225 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 ...

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**2**answers

458 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 ...

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263 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 ...

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180 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 ...

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**0**answers

326 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 ...

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**2**answers

584 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 ...

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**1**answer

280 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 ...

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**5**answers

879 views

### Circumference of convex shapes

Here is a puzzle I found in "Mitteilungen der DMV" (roughly "Letters of the German Society of Mathematicians") issue 19/2011. It was posed by Alfred Schreiber in "Wie man Hasen fangt" (How to catch ...

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**1**answer

377 views

### Example in Guillemin-Sternberg's Convexity Paper

At the end of the introduction of Guillemin-Sternberg's 1982 Inventiones paper "Convexity Properties of the Moment Mapping", there is a nice picture of a non-Abelian moment polytope attributed to ...

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**1**answer

215 views

### Polar interpretation of convexity

Let $C$ be a convex polygon in the plane containing the origin, and let $r(\theta)$ for $\theta\in[0,2\pi)$ be a parametrization of its boundary. Is there a condition on $r$ that is equivalent to (or ...

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2k views

### Minimize trace of inverse of convex combination of matrices.

Hello! (First question--please forgive me if its unclear.)
I am interested in efficient/approximate optimization techniques for minimizing a norm of a convex combination of symmetric, positive ...

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votes

**2**answers

318 views

### Distributing points with respect to a concave function

Suppose I have a concave function defined on the unit interval such that $f(0) = f(1) = 0$ and $\int_0^1 f(t) dt = \alpha$, where $\alpha$ is "small" (say $0.01$ or thereabouts). Say I distribute $n$ ...

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**0**answers

201 views

### How do control the boundary regularity of the Legendre transformation domain from a convex function

Let f(x) be a strongly convex smooth function (its Hessian matrix is positive definite) defined in a convex domain D, introduce the Legendre transformation
$$x=(x_1,...,x_n)\rightarrow ...

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203 views

### Convexity of a constrained optimization problem

Hi, this is a continuation of a previous question I asked about the convexity of an optimization problem I am working with.
Consider the function
\begin{multline}
B_i(a_0,\mathbf{p}) \equiv ...

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**1**answer

324 views

### Convexity of $\frac{1}{2}\vert x_1 + x_2 + x_3 - x_1x_2x_3\vert^2$

Anybody have any tips on how to show that the function $\frac{1}{2}\vert x_1 + x_2 + x_3 - x_1x_2x_3\vert^2$ is convex in $\mathbf{x}$, where $\vert x_i \vert \leq 1$?
This comes from the following ...

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245 views

### Are affine continuous functions on Bauer sub-simplices of the probability measures given by integration over continuous functions?

Let $X$ be a compact (non-metrizable) Hausdorff space and $\mathcal{P}(X)$ the set of Radon probability measures with
weak-$*$ topology (weak topology induced by the continuous functions).
Consider a ...

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votes

**1**answer

253 views

### Integral in a σ−convex set.

Having had no (proper) answer to this question, I formulate the remaining case as a new question as follows. With $I=[0,1]$, let $E$ be a separable (real) Banach space, and let $\gamma:I\to E$ be ...

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514 views

### A lower bound of a particular convex function

Hello,
I suspect this reduces to a homework problem, but I've been a bit hung up on it for the last few hours. I'm trying to minimize the (convex) function $f(x) = 1/x + ax + bx^2$ , where ...

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**3**answers

451 views

### surfaces of constant centro-affine curvature

It is well-known that every closed surface in $\mathbb R^3$ having constant Gauss curvature is a round sphere. Inspired by this question, I'd like to ask whether a similar rigidity holds for ...

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128 views

### Convexity of a cone in CxC

Hi. I have trouble deciding if the set of couples $ \left( \xi, \zeta \right) \in \mathbb{C}^2 $ with $ Re \left( \xi \text{ } \overline{\zeta} \right) > 0 $ is convex. It is a (real) cone, but is ...

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**1**answer

580 views

### volume of the unit ball of the Banach space $\ell_1^n\otimes_{\epsilon}\ell_1^n$?

We denote by $\otimes_{\epsilon}$ the injective Banach tensor product.
Which is the asymptotic volume of the unit ball of the Banach space $\ell_1^n\otimes_{\epsilon}\ell_1^n$?

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**2**answers

451 views

### Existence of extreme points

Let $(X,d)$ be a complete metric linear space whose balls are convex. Let $Y\subseteq X$ be a bounded, closed and convex subset that verifies the following property: for all $y_0\in Y$, the distance ...

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2k views

### Polynomial roots and convexity

A couple of years ago, I came up with the following question, to which I have no
answer to this day. I have asked a few people about this, most of my teachers and some
friends, but noone had ever ...

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vote

**2**answers

489 views

### Convex polynomial homogenization and convexity

I have a polynomial that I know to be convex. If I homogenize the polynomial, is the resulting homogeneous polynomial also convex? I know that the perspective of a convex function is convex, but ...

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**1**answer

362 views

### Approximation of an integral of a concave function

I suspect this is a homework question somewhere, but I've not seen it elsewhere and it seems like it should be easy: let $f(x)$ be a concave function from $[0,1]$ to the reals such that $f(0) = f(1) ...

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1k views

### Infinite convex combinations in a Banach space

Let's say that a subset $C$ of a Banach space $X$ is $\sigma$-convex if the following property holds:
For any sequence $(x_k)_{k\ge0}$ in $C$, and for
any sequence of non-negative real numbers ...

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935 views

### Two-convexity ⇒ Lefschetz?

Assume that
$\Omega$ is an open simply connected set in $\mathbb R^n$
(two-convexity) if 3 faces of a 3-simplex belong to $\Omega$ then whole simplex in $\Omega$.
Is it true that any component of ...

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votes

**2**answers

384 views

### Submanifolds lying on the boundary of a convex domain

Let $M$ be a submanifold of $\mathbb R^n$. Call $M$ locally convex if locally $M$ is contained in the boundary of a convex domain of $\mathbb R^n$.
Is there any known condition that is equivalent to ...

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**0**answers

177 views

### Radon transform and Log-concavity

This question is related to (but different from) that of Darsh Ranjan.
Is there a characterization of the functions $f:\mathbb R^n\rightarrow\mathbb R_{\ge0}$ whose Radon transform $\hat ...

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**0**answers

343 views

### Bounding point-wise maximum of the absolute difference of two convex functions

Let $\Delta: R \times R \rightarrow R_{+}$ be a positive and convex function (convex in, say, both the arguments) called the loss function.
Let $x \in R^d$. Moreover, let $H_1,...,H_r$ be sets of ...

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**1**answer

754 views

### If $f:R^n \to R$ is a smooth real-valued function such that $\nabla f : R^n \to R^n$ is a diffeomorphism, what can one conclude about the behavior of $f(x)$ at infinity?

This question may seem peculiar, so let me preface it by saying that it arose while I was trying to understand Legendre transformations better, and in that context it is fairly natural. Anyway, ...

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**1**answer

575 views

### Does a log-concave function on a convex set extend continuously to the boundary?

Let $U$ be an open convex set in a locally convex space $X$, and let $f : U \to [0,1]$ be a log-concave function on $U$ (i.e., bounded and real-valued). Under what conditions does $f$ have a ...

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387 views

### Does the dual Banach space $B(\ell^\infty)$ have weak* normal structure?

Let $K$ be a bounded closed convex subset of a Banach space $E$. A point $x$ in $K$ is called a diametral point if
$$
\sup_{y\in K} \|x-y\|={\rm diam}(K).
$$
where ${\rm diam}(K)$ denotes the ...

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**2**answers

338 views

### If $K$ and $L$ are compact convex sets with smooth boundary, does their union have piecewise-smooth boundary?

Clarification: by "piecewise", I mean a finite number of pieces.
I'm sure this must be true, but my search for a citation was in vain (although I did learn the new term "polyconvex").
Thanks!

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133 views

### Lower semicontinuity of Bregman distances/divergences

For a Banach space $X$ and a convex functional $J:X \to [0,\infty]$ (i.e. with values in the extended reals), consider the associated Bregman distance: For $x,y\in X$ and $\xi\in\partial J(y)$:
...

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**1**answer

670 views

### Lipschitz properties of minima/minimizers of convex functions of two variables

Suppose I have a function $f(x,y)$ from $\mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ that is convex in both $x$ and $y$. Set
$g(y) = \min_{x} f(x,y)$
What I would like is for $g(y)$ to be ...

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votes

**3**answers

583 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 ...

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**1**answer

271 views

### Can subgradient infer convexity?

It is known that
If $f:U→ R$ is a real-valued convex function defined on a convex open set in the Euclidean space $R^n$, a vector v in that space is called a subgradient at a point $x_0$ in $U$ if for ...

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**1**answer

406 views

### Polygons and mirrors

Given a point $A$ inside a non-convex polygon $P$, is it always possible to place a finite set of mirrors (not necessarily along the boundary of $P$, any position inside $P$ is allowed) given by ...

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**1**answer

285 views

### Convex sets and projections

Hello!
I recently started (it's purely self-education) reading a "Mathematical programming and optimizations" book, did a vast part of the exercises related to the theoretical part and at one moment ...

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**1**answer

586 views

### Orthogonal similarity of matrices

Given $M\in M_n({\mathbb R})$
and $\ell\in{0,\ldots,n-1}$, we define
$$d_\ell(M)=\sum_{j=1}^nm_{j,j+\ell},$$
where the indices are understood mod $n$. In particular, $d_0$ is the trace operator.
Let ...