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4
votes
0answers
155 views

Convex bodies with symmetric shadows.

Theorem. If all orthogonal projections of a convex body $K \subset \mathbb{R}^n$ onto $2$-dimensional subspaces have a center of symmetry, then $K$ has a center of symmetry. This is a classic result ...
0
votes
0answers
72 views

Convexity of a Certain Set of Covariance Matrices

Hello, My question is about a certain set of matrices being convex or not. I'll start with some preliminaries in order to define myself properly. Let $X_1,U,X_2$ be three zero-mean Gaussian random ...
4
votes
1answer
185 views

Extreme rays in the cone of (semi)metrics

How many extreme rays are there on the polytopal cone formed by all semimetrics on a set with $n$ elements? Some background. Given a set $X$ with $n$ elements, the set of all semimetrics $d:X \times ...
0
votes
1answer
132 views

Convex functions: bounding the difference

Suppose you are given a convex function $f: R^d \rightarrow R$. Let us say you are given $x,x' \in R^d$ and $x_1, x_2, \ldots, x_n \in R^d$ such that $$\sum_{i=1}^n (x_i - x') = x - x'.$$ Is it ...
3
votes
1answer
159 views

When are cones of matrices “generated” by vectors?

The cone $P_{n}$ of positive semidefinite matrices of order $n$ can be represented in this form: $P_{n}=\{A|\forall x\geq 0: \langle A,xx^{T}>0 \rangle \}$ with $x$ running over ...
11
votes
2answers
405 views

A problem on convex geometry

Consider a convex body $K \subset \mathbb{R}^n$ containing the origin in its interior. Although the body is not necessarily symmetric, let us say that two points in its boundary $\partial K$ are ...
6
votes
0answers
288 views

Minkowski's Inequality for Integrals in Orlicz spaces

EDIT: I have changed the question to have less parameters, fitting it into the context of Orlicz spaces. Suppose $f:[0,\infty)\to[0,\infty)$ is convex and increasing, ...
11
votes
1answer
345 views

Convex cones and self-duality

Consider the Euclidian space $E_n={\mathbb R}^n$, with standard scalar product $$x\cdot y=x_1y_1+\cdots+x_ny_n.$$ A closed convex cone $\Gamma\subset E_n$ defines an order by $y\ge x$ iff ...
2
votes
2answers
308 views

Convex upper bound on a linear-fractional function

I have a function of the form $f(x,y) = \frac{x}{c+y}$ where $c$ is a positive constant, $c \ge x \ge 0$, and $y \ge 0$. I would like to find a convex upper-bound for this function. Is there a ...
1
vote
1answer
132 views

Smallest Lipschitz constant on non-convex domains

It is well known that if a function $f:U\to \mathbb C^n$, $U\subset \mathbb C^m$ satisfies $\sup_{x\in U}\|Df(x)\|_{\infty} = C < \infty$ uniformly on $U$ and $U$ is compact and convex, then $f$ is ...
2
votes
3answers
386 views

A consequence of convexity

Let $f:\mathbb{R}\to\mathbb{R}$ a convex decreasing function. Let $x_0 < x_1 < x_2$. Studying the behaviour of the difference quotient, it is clear that $$f(x_0)-f(x_2) \leq M (f(x_0)-f(x_1))$$ ...
9
votes
2answers
212 views

Can you see smoothness of the boundary of a convex body from its shadow?

Let $d \geq 3$ and suppose that $K \subset \mathbb{R}^{d}$ is a convex body (compact, convex, non-empty interior). Is the following true? The boundary $\partial K$ is a $C^1$-manifold if and only ...
3
votes
2answers
344 views

Convexity in $\{0,1\}^n$

how is convexity defined in a subset $A \subset \{0,1\}^n$? furthermore, is there any extention of the Brunn-Minkowski inequality for subsets of $\{0,1\}^n$? thanks. Edit (previously posted as an ...
0
votes
0answers
179 views

Geometric Mean of Positive Matrices

Hello all, My question regards the geometric mean (GM) of two positive matrices. The definition of the GM for two positive matrices $(A,B)$ is given by: ...
0
votes
1answer
140 views

necessary and sufficient conditions for a function to be DC

Hi, Does anyone know the necessary and sufficient conditions for a function to be a DC-function? Definition: A function is a DC-function if and only if it can be written as a differnece of 2 convex ...
2
votes
1answer
132 views

When a semigroup extension $S \times A$ of a semigroup $S$ with a commutative group $A$ contains a copy of $S$?

Maybe is a trivial question, but I don't know how to handle it. Setting: Let $S$ be a semigroup (i.e. has an associative operation with neutral element $e$) and let $(A,+)$ be a commutative group ...
8
votes
1answer
539 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 ...
2
votes
0answers
127 views

A subclass of log-concave functions satifying a sum inequality

Suppose $f:[0,\infty)\to[0,\infty)$ is continuous and for all positive integers $m$ and real $x\geq0$, $\alpha,\beta>0$: $$ ...
6
votes
0answers
193 views

Is $Q_n(x)=\sigma_{n+1}(x)/\sigma_n(x)$ logarithmically convex on $\mathbf{R}$?

In 1975 J. van de Lune considered the monotony properties of the canonical Riemann Upper and Lower sums for $\int_0^1 t^xdt$, with $x>0$. Writing $\sigma_n(x) := 1^x+2^x+\cdots+n^x$ these sums are ...
7
votes
2answers
266 views

Duality between extremal points and extremal maps

Suppose I have a convex set $C\subset\mathbb{R}^n$ such that $0\in C$ and every Cauchy sequence in $C$ converges in $C$, but $C$ need not be bounded. (Actually I want unbounded $C$). Consider the set ...
0
votes
1answer
101 views

Is these two optimization problems share the same solution?

Hello all, I am dealing with some SDP optimization, and I come across the following problem. The optimization problem is given by \begin{aligned} &\operatorname*{min}_{t_1,\ldots,t_m,X}\ \sum ...
6
votes
1answer
3k views

Eigenvalues of product of two symmetric matrices

This is mostly a reference request, as this must be well known! Let $A$ and $B$ be two real symmetric matrices, one of which is positive definite. Then it is easy to see that the product $AB$ (or ...
1
vote
1answer
268 views

Question regard checking convexity by “restriction to any line that intersects the function domain”

Hello all, I have a question (probably stupid one) about the fact that " A function is convex if and only if it is convex when restricted to any line that intersects its domain". In Stephen Boyd and ...
0
votes
1answer
725 views

Is a jointly convex function of x and y convex as a function of x when y=z(x)?

Hi, Suppose that $x \in R^m, y \in R^n, z(x) \in R^n$, and $f(x,y)$ is convex in $(x,y)$. Is $f(x,z(x))$ a convex function in $x$ for arbitrary continuous functions $z(x)$? Thanks!
1
vote
2answers
278 views

non convex optimization

Hi there, In my studies I come up with this nonconvex optimization problem argmin |Ax|_2+lamda*|x|_1 subject to x'x=1 where cost function is nonsmooth but convex and the constrant in nonconvex. I ...
0
votes
1answer
157 views

(probably simple) optimization question

Suppose you have a concave function defined over a non-polyhedral convex cone and you are interested in the infimum. What would be standard approaches to tackle the question? (The cone is actually PSD ...
2
votes
1answer
303 views

Proving that a specific function is quasiconvex

Hello all, Assume we have a sequence of quasiconcave functions (in $X$) denoted by $f_{i,j}(X)$ for $i,j = 1,\ldots,n$. Denote by $F(X)$ the $n\times n$ matrix whose $(i,j)$ entry is the function ...
1
vote
1answer
254 views

Conjecture that two nested convex curves have a point with the same slope

I'm trying to prove a conjecture and need some help. Consider a continuous, twice differentiable function $p(a)$ such that $p(0) = 0$ and $\forall a$, $p'(a) > 0$ and $p''(a) < 0$ and $p$ is ...
3
votes
1answer
167 views

Are faces of a compact, convex body “opposed” iff their extreme points are pairwise “opposed”?

Let $P$ be a compact, convex subset of $\mathbb{R}^n$ (infinite-dimensional generalisations welcome, but not necessary). Let's say that disjoint subsets $W_1$, $W_2$ $\subset P$ are opposed if there ...
7
votes
3answers
266 views

Characterising semi-definite positiveness on vectors with non-negative entries

My problem is to characterise (or find useful information on) the cone $C$ of $N\times N$ matrices $M$ ($N\geq 1$) such that $$V^t M V\geq 0$$ for every vector $V $ with non-negative entries. Is this ...
20
votes
1answer
1k views

A Weak Form of Borsuk's Conjecture

Problem: Let P be a d-dimensional polytope with n facets. Is it always true that P can be covered by n sets of smaller diameter? Background and motivation The Borsuk conjecture (disproved in ...
4
votes
1answer
273 views

Going in the direction of the gradient

First, a motivating example. Suppose $f(x)$ is convex, differentiable, with a single minimum $x^*$. Then the differential equation $$\dot{x}(t) = -\nabla f(x(t))$$ drives $x(t)$ to $x^*$. Now my ...
1
vote
1answer
508 views

proving the interior of a dual cone is the set of vectors whose inner product is strictly positive on the cone

Apologies for posting such a simple question to mathoverflow. I've have been stuck trying to solve this problem for some time and have posted this same query to math.stackexchange (but have received ...
6
votes
4answers
381 views

Packing and isoperimetrics

Suppose a manufacturer bottles small units of liquid and ships them via very large trucks. If the transportation cost nothing, spherical bottles would minimize the packaging cost (isoperimetric ...
5
votes
1answer
213 views

Convexity of a specific semialgebraic set

I have an engineering problem which maybe resolved with semi-definite programming optimization. I have a set which I would like to know if is convex: Being $m \in \mathbb{R}^+$ a positive real ...
4
votes
1answer
269 views

Deciding the convexity of semialgebraic sets

Given a basic closed semialgebraic set, $S \subset \mathbb{R}^n$, defined by $S = \{ x \in \mathbb{R}^n \mid g_1 (x) \geq 0 \land \dots \land g_m (x) \geq 0\}$ where $m \in \mathbb{N}$ and $g_1, ...
4
votes
4answers
623 views

Is the intersection of boundaries of convex bodies a topological sphere?

Let $K_1, K_2, \ldots K_n$ be convex bodies in $R^d$. Assume that for any index set $I$, $\cap_{i \in I} K_i$ is not empty and is not properly contained in any body $K_i$ for $i\in I$. Is it true ...
13
votes
2answers
454 views

A natural center of a convex weakly compact set in Banach space

Question: Let $S$ be a convex weakly compact set in Banach space $H$. Propose a natural way to define the unique center $O \in S$. Motivation: A lot! For example, in game theory $S$ can be a set of ...
2
votes
1answer
277 views

Quermassintegrals as mean curvature integrals

It is well-known that the quermassintegrals $W_{i}$ of a convex body $K\subset \mathbb{R}^{n}$ can be written as a mean curvature integral $$ W_{i}(K)=n^{-1}\int_{\partial K}H_{i-1} ...
2
votes
2answers
392 views

Does the minima of a sequence of convex convergent functions converge?

Suppose $f_1,f_2,\ldots $ is a sequence of convex functions that converges to a continuous convex $f$. Let $x_1^*,x_2^*$ be their respective (not necessarily unique) minima, and let y be a minima of ...
2
votes
1answer
155 views

Lipshitz Constant of the convex extension of a submodular function

The title says it all :) Given a submodular function (take the rank function of a matroid, for a concrete example) $f:\{0,1\}^n\rightarrow \mathbb{R}$, we can extend it to a convex function ...
0
votes
0answers
220 views

A question regarding Danskin's theorem

Suppose $\phi({\bf x},{\bf z})$ is a continuous function of two arguments, $\phi: {\mathbb R}^N \times Z \rightarrow {\mathbb R}$, where $Z \subset {\mathbb R}^m$ is a (non-empty) compact convex ...
5
votes
1answer
334 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
1answer
116 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
1answer
315 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
3answers
1k 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
1answer
305 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: ...
2
votes
1answer
220 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
0answers
168 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 ...