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2
votes
0answers
140 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
2answers
204 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
1answer
259 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
1answer
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 ...
2
votes
2answers
508 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
1answer
841 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
1answer
156 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
2answers
441 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 ...
7
votes
0answers
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 ...
3
votes
2answers
455 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
0answers
259 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
0answers
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 ...
0
votes
0answers
323 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 ...
13
votes
2answers
580 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
1answer
278 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 ...
2
votes
5answers
867 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 ...
3
votes
1answer
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 ...
3
votes
1answer
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 ...
4
votes
3answers
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 ...
5
votes
2answers
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 $n$ ...
0
votes
0answers
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 ...
1
vote
0answers
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 ...
4
votes
1answer
323 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 ...
2
votes
0answers
243 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 ...
0
votes
1answer
250 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 ...
3
votes
2answers
509 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 ...
8
votes
3answers
450 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 ...
0
votes
0answers
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 ...
8
votes
1answer
576 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 $\ell_1^n\otimes_{\epsilon}\ell_1^n$?
2
votes
2answers
448 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 ...
36
votes
4answers
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 ...
1
vote
2answers
486 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 ...
6
votes
1answer
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) ...
12
votes
0answers
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 ...
30
votes
0answers
930 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 ...
3
votes
2answers
382 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 ...
2
votes
0answers
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 ...
1
vote
0answers
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 ...
17
votes
1answer
749 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, ...
5
votes
1answer
573 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 ...
1
vote
0answers
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 ...
2
votes
2answers
337 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!
1
vote
0answers
131 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 $\xi\in\partial J(y)$: ...
3
votes
1answer
659 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 ...
3
votes
3answers
580 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
1answer
270 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 ...
5
votes
1answer
404 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 ...
0
votes
1answer
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 ...
5
votes
1answer
579 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 ...
0
votes
3answers
531 views

a different algebra/representation for convex sets

Hi, I was dealing with finding a feasible region for a set of norm inequalities and the feasible region is convex. The question is not about how to find the feasible region but how to represent the ...