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0
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0answers
95 views

Uniform convergence of difference quotients of a convex function

Let $f(\cdot):\mathbb{R}^n\rightarrow \mathbb{R}$ be a convex function. At $x$ denote the subdifferential by $\partial f(x)$ which is compact and closed. Now, define the approximation of $f$ around a ...
0
votes
1answer
511 views

Is a convex function continuous and almost everywhere differentiable? [closed]

is this statement true ? assume $f:D\rightarrow \mathbf{R} $ is a convex function where $D\subset \mathbf{R}^n$ is a convex set. $f$ is continuous and almost everywhere differentiable and in class ...
1
vote
1answer
178 views

How to examine the convexity of a complex function numerically?

I have a function which does not have a closed form . Large numerical effort should be done to evaluate the function for even a single point. How can I examine the convexity of my function over the ...
0
votes
1answer
170 views

carleman inequality

Is there a connection between carleman inequality discovred by T. Carleman in 1922 if I am not mistaken in his research on quasianalytic functions and what is called Carleman estimates used in the PDE ...
2
votes
0answers
97 views

Why pseudoconvexity is important in Partial differential equation theory?

I am a new researcher in mathematics and I work on convexity. Are convexity and pseudoconvexity related topics and in which respect to PDE theory ? One of the important results in PDE theory is the ...
1
vote
2answers
189 views

Construction of a convex function nondifferentiable on a countable set [closed]

Let $H$ be a countable subset of $[0,1]$. Construct a convex function $f:[0,1]\rightarrow\mathbb{R}$ such that $f$ is nondifferentiable on $H$ and differentiable in the rest.
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2answers
134 views

Convexity of a minimum function

I was reading a proof of $9g-9$ theorem which states that $9g-9$ length parameters are sufficient the parametrize the Teichmuller space of a closed surface of genus $g$. The proof uses the following ...
4
votes
0answers
110 views

Hilbert metric of a sum of cones

Suppose that $K_{1}$ and $K_{2}$ are pointed closed cones in a finite-dimensional space $V$ whose Hilbert metrics $d_{1},d_{2}$ are known. Is there a way to express the Hilbert metric of $K_{1}+K_{2}$ ...
1
vote
1answer
131 views

A question concerning convex functions

Consider points $a,b,c$ (not on a line) and $x_1,...,x_n$ in $\Bbb{R}^2$. I am looking for a necessary and sufficient condition in terms of the geometric configuration of these points such that for ...
8
votes
1answer
409 views

Small quadrilaterals containing a given convex region

It is easy to prove that (*) Every convex planar set of area 1 is contained in a quadrilateral of area 2. It is also easy to see that statement (*) remains true if the constant 2 is replaced with a ...
1
vote
0answers
53 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
90 views

Linearization of cones

Suppose that $K$ is a closed convex cone in $R^{n}$. Is there a "nice" function $f:R^{n} \rightarrow R^{m}$ so that $f(K)$ is a subspace? What about an approximate subspace?
1
vote
0answers
57 views

Reference request: Compact metrizable semilattices with small connected semilattices are absolute retracts

Let $X$ be a compact metrizable topological semilattice with a neighbourhood base of sub-semilattices that are path connected. I need a reference for a proof that $X$ is an absolute retract. Here is ...
1
vote
1answer
148 views

Helly's number from biconvex functions

Helly's Theorem states the following. Suppose $X_1,X_2,...,X_N$ are convex sets in $\mathbb{R}^d$, such that for any index-set $I$ with $|I| \leq h(d) := d+1$, we have $\bigcap_{i \in I} X_i \neq ...
0
votes
0answers
49 views

convexity of two linear spaces connected by convex nonlinear equality constraints

If there are two sets of linear constraints in different variables, Ax <= b with x_l <= x <= x_u and Cy <= d with y_l <= y <= y_u, and a set of equality constraints of a specific ...
4
votes
1answer
175 views

Zoll Flat Finsler tori and convex bodies on a starry night

The starry night. The "celestial sphere" is given by set of non-zero vectors in $\mathbb{R}^n$ modulo positive dilations (i.e., $v \equiv w$ if $v = \lambda w$ for some $ \lambda > 0$) and the ...
1
vote
2answers
206 views

On a version of gradient descent

I am trying to read this paper and have gotten stuck. The author considers the problem of minimizing a convex function whose gradient has Lipschitz constant $M$ and considers the scheme $$ x(t+1) = ...
1
vote
1answer
75 views

Constrained minimum maximal distance.

Let $C$ and $D$ be two convex sets. And suppose $C\cap D\neq \emptyset$. Let $x^*$ is the solution to the optimization problem: $$\min_{x\in C} \max_{y \in D} |x-y|^2$$ Is it true that $x^* \in D$. ...
0
votes
2answers
232 views

Higher dimensional convex hull

Let $CH(S)$ be a convex hull of a finite set $S$ and denote the set of all the vertices of $CH(S)$ as $Vert(S)$. For a vertex $v \in Vert(S)$, it has an associated set $E(v)$ which is defined as ...
1
vote
1answer
145 views

Regularizing a Convex function with itself

Hi, This is a problem that has being bothering me the last few days. Assume a convex function $f(x): {\mathbb R}^n \rightarrow {\mathbb R}$ with a unique minimizer $x^{\star}$. Now consider the ...
2
votes
3answers
221 views

Measuring the distance of a convex set from a ball (Nikodym distance)

Assume we have a centrally symmetric convex set $K \subset \mathbb{R}^n$ such that Vol(K)=1. In addition, assume that for every direction $u$ we know that $Vol(K \Delta R_u(K)) < \epsilon$, where ...
1
vote
0answers
59 views

Which matrix/operator in a cone has the largest negative spectral part?

Background: Let $\mathcal{K}$ be set (convex cone, if you like) of symmetric matrices of order $n$. Each matrix $A \in \mathcal{K}$ can be decomposed in a unique way as $A=A_{+}-A_{-}$, where ...
5
votes
1answer
276 views

A question on the Mahler conjecture

In its asymmetric version, the Mahler conjecture states that if $K \subset \mathbb{R^n}$ is a convex body containing the origin as an interior point and $$ K^* := \{y \in \mathbb{R}^n : \langle y, x ...
1
vote
1answer
224 views

When does the finite union of convex sets have a hole in it?

Let $f_1, \dots, f_j$ be convex functions from $\mathbb{R}^n \to \mathbb{R}$. I am trying to develop a test that decides whether or not the set $\{x | f_1(x) \le k_1\} \cup \dots \cup \{x | f_n(x) ...
3
votes
1answer
99 views

What is the doubling dimension of convex functions?

I am interested in the complexity of convex functions, specifically the "doubling dimension" of the class of convex functions defined on a compact subset of Euclidean space, when compared using the ...
3
votes
2answers
332 views

Is Ryser's conjecture on permanent minimizers still open?

Let $A(k,n)$ be the set of $\{0,1\}$ matrices of order $n$ with all their line sums equal to $k$. Conjecture number 5 on the list from Minc's book, attributed to Ryser, says that if $A(k,n)$ ...
3
votes
0answers
77 views

Covering points with a convex hull

Consider a set of $n$ points $x_1,\ldots,x_n \in \mathbb{R}^d$, for some $n \gg d$. Suppose $\{x_1,\ldots,x_n\} \subset C \subset \mathbb{R}^d$. Say that a set of points $y_1,\ldots,y_m \in C$ covers ...
0
votes
1answer
183 views

Extensions of Carathéodory's theorem

We know about the Carathéodory's theorem which is on the convex bodies of $\mathbb{R}^d$. My question is, how far we can extend it? Is it true for say, any convex object of Banach space, or for convex ...
4
votes
1answer
170 views

Monotonicity of Loewner ellipsoid?

Given two $0$-symmetric convex bodies $K \subset L \subset \mathbb{R}^n$, is it true that the Loewner ellipsoid of $K$ is contained in the Loewner ellipsoid of $L$? I have just finished a proving a ...
3
votes
1answer
372 views

Why are all these families of polynomials finally log-concave?

This started when I was examining certain families of unimodal polynomials, i.e. $\sum_{k=0}^n a_kx^k$ where $a_0\le a_1\le\cdots \le a_k\ge\cdots \ge a_n$. (Notation: in the following, the $a_k$ ...
3
votes
1answer
259 views

Possible to find a set of log-concave functions with log-concave sums?

While the set of log-convex functions is closed under addition, the set of log-concave functions is not. Yet if $f$ is log-concave, $\ln(k f) = \ln(k)+\ln(f)$, with $k \in \mathbb{R}^+$ constant, is ...
2
votes
0answers
35 views

Conditions under which a set of points have a low weight representation under some basis

Let $S \subset R^d$ be a convex polytope in $d$ dimensional real space. Say that $S$ has weight $w$ if there exists some basis $x_1,\ldots,x_d$ such that for every point $v \in S$, we can write $v = ...
6
votes
2answers
141 views

On the convexity of element-wise norm 1 of the inverse

Question first asked on math.stackexchange here: http://math.stackexchange.com/questions/317209/on-the-convexity-of-element-wise-norm-1-of-the-inverse On the convexity of element-wise norm 1 of the ...
1
vote
0answers
64 views

Fastest 'Oracle' Algorithm for satisfying a single linear constraint on a convex set?

In this paper by Arora, Hazan, and Kale (http://www.cs.princeton.edu/~arora/pubs/MWsurvey.pdf) a method is given for using the Multiplicative Weights Update algorithm to quickly solve Linear Programs ...
5
votes
2answers
274 views

How do you prove that every curve of constant width is convex?

Cross-posted from math.stackexchange: Let C be a simple closed plane curve and let D be its interior. Recall that the width of C in a direction θ is the distance between two supporting lines for D ...
3
votes
1answer
132 views

Support Functions Of 3D Convex Bodies In Spherical Polar Coordinates

What are the known sufficient conditions,analagous to the planar curvature condition, in terms of functions of theta and phi, on the support function h(theta,phi) of a surface in 3D which imply it is ...
1
vote
1answer
202 views

Covering the cone of positive semidefinite matrices by intervals

Is it possible to cover the cone of positive semidefinite matrices by a finite/countable/interesting family of closed intervals of matrices? How about a general convex cone? For the finite case the ...
4
votes
5answers
280 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 ...
2
votes
1answer
216 views

another diameter-perimeter-area inequality

Recently I learnt that $$ \inf\frac{diam(C)(per(C)-2diam(C))}{area(C)}=0$$ where the infimum is taken over all plane convex bodies $C$ (say, with non-zero area). In other words, there is no ...
3
votes
2answers
262 views

a diameter-perimeter-area inequality for convex figures

Is the following inequality known? I believe it's true, but I could find no reference. For any convex body $C$ in the plane we have $$\left(4-\frac{8}{\pi}\right)area(C)\leq > ...
4
votes
0answers
154 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
180 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
130 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
155 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
403 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
284 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
334 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
302 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
129 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 ...