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3
votes
1answer
63 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
0answers
55 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 $$ ...
0
votes
0answers
35 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
1answer
73 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, ...
15
votes
1answer
565 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
votes
0answers
49 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 ...
0
votes
1answer
120 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}}}} \\ ...
7
votes
1answer
197 views

In what sense is the Bayesian posterior mean a “convex combination”?

I asked this on math.stackexchange with no response, I'm hoping someone here might have something. Suppose I want to estimate $x \in \mathbb{R}^n$ from two signals with zero mean, normally ...
4
votes
1answer
181 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 ...
4
votes
7answers
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 ...
2
votes
0answers
29 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 ...
4
votes
6answers
382 views

Characterizing Convex Configurations of Quadrupels of Coplanar Points via Linear (In-)equalities between Distance Sums or Differences

Given 4 points $A$, $B$, $C$ and $D$ in general position in the euclidean plane, is it possible to determine from the 6 distances $AB$, $BC$, $CD$, $AD$, $AC$ and, $BD$ alone, whether every point is a ...
1
vote
0answers
55 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 ...
2
votes
1answer
97 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
1answer
113 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
0answers
150 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
2answers
103 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
1answer
39 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
3answers
258 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 ...
3
votes
6answers
1k 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 ...
2
votes
1answer
113 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 = ...
1
vote
0answers
40 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 ...
10
votes
2answers
650 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 $\ell_1^n\otimes_{\epsilon}\ell_1^n$?
1
vote
0answers
27 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 ...
3
votes
1answer
505 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
1answer
249 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 $n$?
11
votes
1answer
299 views

Choquet theory and Hilbert's fourth problem

The following text is an attempt to see Hilbert's fourth problem in a new light. Definition. A pseudometric $d$ on $\mathbb{R}^n$ is called projective if whenever a point $z$ belongs to a line ...
14
votes
1answer
2k 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 ...
2
votes
0answers
97 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 ...
1
vote
1answer
214 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 ...
7
votes
1answer
249 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 ...
3
votes
1answer
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\ $ ...
5
votes
2answers
432 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 ...
51
votes
2answers
845 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 ...
18
votes
1answer
844 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
3answers
484 views

Convex hulls of families of probability measures

Let $X$ be a standard Borel space, so that the space of Borel probability measures on $X$ is also a standard Borel space. We denote it by $\mathcal P(X)$. In this paper for any family of probability ...
8
votes
2answers
220 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 ...
6
votes
0answers
252 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 ...
4
votes
1answer
199 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
2answers
261 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 ...
13
votes
0answers
267 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
1answer
90 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 ...
3
votes
1answer
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 ...
11
votes
2answers
927 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
0answers
106 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 ...
1
vote
0answers
91 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 ...
6
votes
0answers
92 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
1answer
319 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
0answers
50 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 ...
5
votes
1answer
83 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 ...